RESISTANCE SWITCHING MECHANISM IN TIO2

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF MATERIALS SCIENCE AND

ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Seong Geon Park

May 2011

This dissertation is online at: http://purl.stanford.edu/cd568rw1925

© 2011 by Seong Geon Park. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

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I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Yoshio Nishi, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Bruce Clemens, Co-Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Blanka Magyari-kope

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

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ABSTRACT

Resistive Random Access Memory (ReRAM) has attracted significant attention recently,

as it is now considered as the promising candidate for the next generation of non-volatile

memory device, due to its high density, low operating power, fast switching speed, and

compatibility with conventional CMOS process. Among many resistance switching

materials, TiO2 has been widely studied. However, the most challenging issue is that the

underlying switching mechanism is lacking in-depth understanding.

It has been proposed that the resistance switching is strongly coupled with the

presence and a preferential distribution of oxygen vacancies involving the formation of a

conductive filament. Although many experiments have been done to address the

switching mechanism during the last decade, it is hard to figure out what happens at

microscopic level. Therefore, systematic interpretation about the microscopic details of

the role of oxygen vacancies in the formation of a conductive filament is essential. To

address the conduction and the resistance switching mechanism, the effect of oxygen

vacancies on the electronic structures in TiO2 has been investigated using first principles

calculations based on density functional theory.

In this dissertation, we report “ON”-state (Low Resistance State) conduction

mechanism of rutile TiO2 including oxygen vacancies, and then the transition from “ON”

to “OFF”-state (High Resistance State) is investigated. Although it is known that TiO2

exhibits n-type semiconducting property with extra electrons generated by the formation

of oxygen vacancies, “ON” and “OFF”-state conductivity during resistance switching

cannot be explained by isolated single oxygen vacancy.

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We calculated electronic characteristics such as density of states, electron

localization function, band decomposed charge density distribution, and energy band

structure, and show the influence of oxygen vacancy configurations on these properties

and on the resistance change. Oxygen vacancy ordering and diffusion of either oxygen

vacancy or hydrogen impurities have a significant impact on both the formation of the

conductive filament and the transition from “ON” to “OFF”-state. Results from this study

indicate that the “ON”-state conduction and resistance switching model that can be

ascribed to the formation and rupture of conductive filament consisting of oxygen

vacancy-ordered structure.

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ACKNOWLEDGMENTS

In the first place, I would like to record my foremost gratitude to my advisor Yoshio

Nishi for his supervision, advice, and guidance from the very early stage of this research

as well as giving me extraordinary experience throughout the work. He provided me

unflinching encouragement and support in various ways, and gave me the freedom to

explore different ideas of research. I am indebted to him more than he knows.

I gratefully thank Blanka Magyari-Köpe. She helped me so much to do all

simulation works during my Ph.D, and teach me everything about ab initio simulation

works. I also would like to thank Professor Bruce M. Clemens for being a member of my

reading committee. I deeply appreciate advice and inputs on my studies and dissertation.

I thank Professor Xiaolin Zheng for being the defense chair and Professor Mark

Brongersma for being on my defense committee despite their busy schedules.

Many thanks go in particular to Baylor Triplett and Michael Deal. I am much

indebted to them for their valuable advice and discussion on the resistive switching

mechanism in ReRAM devices. With their help, I was able to make an improvement in

the research.

My research project was initially sponsored by Non-volatile Memory Technology

Research Initiative (NMTRI) which is funded by Intel, Micron, Samsung, Toshiba,

Applied Materials, SanDisk, Texas Instruments, and Spansion. I would like to thank all

company members who gave me valuable comments at NMTRI review meeting. I also

wish to thank all former and current members of Nishi group. Their help and friendship

made my graduate school experience more enjoyable. Many thanks also to Sandy and

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Gabrielle for assisting me through all problems I encountered while conducting the

research.

I would also acknowledge Hyung Dong Lee, Jae Yun Yi, and Seung Wook Ryu

for their advice and their willingness to share their bright thoughts with me, which were

very fruitful for shaping up my ideas and research.

During the past five years, I have made many good friends in the campus and in

the Escondido Village. Without their physical and mental support to my family and me, I

would not be able to finish my Ph.D work. I would like to thank all of these friends. I

offer special thanks to Korean MSE students.

Finally, I would like to express my deepest gratefulness and love to my wife,

Kyunghwa and my son, Jaemin. Their continuous support, love and sacrifice allowed me

to pursue my dream and made this dissertation possible. I dedicate this dissertation to my

family.

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TABLE OF CONTENTS

Abstract .......................................................................................................................................... iv

Ackowledgments ............................................................................................................................ vi

List of Tables .................................................................................................................................. xi

List of Figures ................................................................................................................................ xii

CHAPTER 1: INTRODUCTION ............................................................................................................ 1

1.1 MOTIVATION ........................................................................................................................ 2

1.2 BASIC OPERATIONAL PRINCIPLES OF RESISTANCE SWITCHING .......................................... 3

1.3 CLASSIFICATION OF THE RESISTANCE SWITCHING MECHANISMS ...................................... 4

1.4 SWITCHING MATERIALS AND TITANIUM DIOXIDE............................................................... 7

1.5 OUTLINE OF DISSERTATIOIN ................................................................................................ 9

REFERENCES .............................................................................................................................. 11

CHAPTER 2: Computational Methodology ..................................................................................... 14

2.1 COMPUTATIONAL DETAILS ................................................................................................ 14

2.2 LIMITATIONS OF LOCAL DENSITY APPROXIMATION (LDA) .............................................. 15

2.2.1 UNDERESTIMATION OF ENERGY BAND GAP ............................................................... 15

2.2.2 WRONG DESCRIPTION OF DEFECT STATES IN THE ENERGY BAND GAP .................... 17

2.3 EFFECT OF ON-SITE COULOMB CORRECTION (U) .............................................................. 18

2.3.1 ELECTRONIC STRUCTURE OF RUTILE TIO2 ................................................................. 18

2.3.2 U VALUE FROM PREVIOUS LITERATURE .................................................................... 20

2.3.3 EFFECT OF Ud IN LDA+U ............................................................................................ 21

2.3.4 EFFECT OF UP IN LDA+U ............................................................................................ 24

REFERENCES .............................................................................................................................. 27

CHAPTER 3: SINGLE OXYGEN VACANCY ....................................................................................... 32

3.1 EFFECT OF U ON DEFECT STATES ...................................................................................... 33

3.1.1 POSITION OF DEFECT STATES IN DENSITY OF STATES ............................................... 33

3.1.2 PARTIAL DENSITY OF STATES OF NEIGHBORING TI ................................................... 35

3.2 ELECTRON LOCALIZATION FUNCTION ............................................................................... 37

3.2.1 ELECTRON LOCALIZATION ON TI ................................................................................ 37

3.2.2 ATOMIC RELAXATIOIN AROUND OXYGEN VACANCY ................................................ 39

3.3 TOTAL DENSITY OF STATES AND CHARGE DENSITY DISTRIBUTION ................................. 41

ix

3.3.1 DEEP LEVEL DEFECT STATE OF AN ISOLATED SINGLE VACANCY ............................. 41

3.3.2 CHARGE DENSITY DISTRIBUTION ............................................................................... 43

3.4 OXYGEN VACANCY FORMATION ENERGY ......................................................................... 46

3.5 POSITIVELY CHARGED OXYGEN VACANCY ....................................................................... 49

3.5.1 DENSITY OF STATES .................................................................................................... 49

3.5.2 ELECTRON LOCALIZATION AND ATOMIC RELAXATION EFFECTS .............................. 50

3.5.3 STABILITY OF CHARGED OXYGEN VACANCY ............................................................ 51

REFERENCES .............................................................................................................................. 54

CHAPTER 4: DI-VACANCY ............................................................................................................. 57

4.1 INTERATOMIC DISTANCE EFFECTS FOR DI-VACANCY ....................................................... 57

4.1.1 STABLE DI-VACANCY CONFIGURATION ..................................................................... 58

4.1.2 INTERACTION BETWEEN TWO VACANCIES IN THE (110) PLANE ................................ 60

4.2 TI-TI BONDS IN EQUATORIAL DI-VACANCY...................................................................... 63

4.3 CHARGE OCCUPANCY OF THE t2g ORBITAL OF TI .............................................................. 67

4.3.1 PARTIAL CHARGE DENSITY AND ORBITAL OCCUPANCY ........................................... 67

4.3.2 PREFERENTIAL DISTRIBUTION OF EXCESS ELECTRONS ............................................. 69

REFERENCES .............................................................................................................................. 72

CHAPTER 5: EFFECT OF VACANCY ORDERING ON THE “ON”-STATE CONDUCTION ................... 74

5.1 MULTI VACANCY CONFIGURATIONS ................................................................................. 75

5.2 FORMATION OF A CONDUCTIVE CHANNEL BY OXYGEN VACANCY ORDERING ................ 76

5.2.1 DENSITY OF STATE OF THE VACANCY-ORDERED STRUCTURE .................................. 76

5.2.2 CHARGE DENSITY DISTRIBUTION AND BADER ANALYSIS ......................................... 78

5.3 STABILITY OF THE VACANCY-ORDERED STRUCTURE ....................................................... 82

5.4 ELECTRONIC BAND STRUCTURE ........................................................................................ 84

5.5 OXYGEN VACANCY ORDERING IN MAGNÉ LI PHASE ......................................................... 88

REFERENCES .............................................................................................................................. 92

CHAPTER 6: TRANSITION FROM “ON”-STATE TO “OFF”-STATE ................................................... 94

6.1 RUPTURE OF THE CONDUCTIVE CHANNEL ........................................................................ 95

6.2 DIFFUSION OF OXYGEN VACANCY .................................................................................... 96

6.3 EFFECT OF HYDROGEN IMPURITIES ................................................................................. 100

6.3.1 INTERSTITIAL HYDROGEN ....................................................................................... 101

6.3.2 HYDROGEN-VACANCY COMPLEXES ........................................................................ 103

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6.3.3 RESET SWITCHING BY HYDROGEN .......................................................................... 104

6.4 PROPOSED RESISTANCE SWITCHING MODEL ................................................................... 106

REFERENCES ............................................................................................................................ 109

CHAPTER 7: CONCLUSION ............................................................................................................ 112

7.1 CONTRIBUTIONS ............................................................................................................... 113

7.2 FUTURE WORKS ............................................................................................................... 115

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LIST OF TABLES

Table 1-1: Summary of resistance switching materials. .................................................................. 7

Table 2-1: Summary of structural parameters a(Å ), c(Å ), bulk modulus B (GPa), and band gap

energy Eg (eV) calculated by LDA+Ud with experimental results. ................................. 22

Table 2-2: Summary of structural parameters a(Å ), c(Å ), bulk modulus B (GPa), and band gap

energy Eg (eV) calculated by LDA+Ud+U. ..................................................................... 25

Table 3-1: Summary of theoretical results of oxygen vacancy formation energy in rutile TiO2. .. 47

Table 3-2: Summary of formation energy of oxygen vacancy with various values of Ud and U

p. 48

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LIST OF FIGURES

Fig. 1-1: Schematic diagram of two different resistance switching modes. (a) unipolar, and (b)

bipolar. (based on Ref. [15]). ............................................................................................ 4

Fig. 1-2: Classification of resistive memory technologies according to their primary principle of

operation (based on ITRS 2010 updates). ......................................................................... 5

Fig. 1-3: X-ray fluorescence mapping after electroforming in Cr-doped SrTiO3 showing the

distribution of oxygen vacancies between two electrodes. [16]………………………….6

Fig. 1-4: (a) High resolution TEM image of a nanofilament. (b) Local I-V curves measured on

nanofilament and TiO2 in background…………………………………………………...8

Fig. 2-1: Unit cell structure of rutile TiO2. Small light blue ball denotes a Ti atom and big red one

an O atom. Each Ti atom is surrounded by six O atoms……….……………………….19

Fig. 2-2: Partial density of states of Ti 3d orbital (solid line) and O 2p orbital (dotted line)

calculated with LDA. The top of the valence band is set to 0 eV…………..…………..20

Fig. 2-3: Density of states obtained by (a) LDA+Ud, and (b) LDA+U

d+U

p. The top of the valence

band is set to 0 eV. Ud and U

p range from 3 to 9 eV and U

d is fixed to 8 eV in the case of

(b)……………………………………………………………..………...………………23

Fig. 2-4: Energy band structure and density of states calculated by (a) LDA+Ud and (b)

LDA+Ud+U

p. Fermi level is not scaled to 0 eV on the energy scale….………………..24

Fig. 3-1: Schematic structure of isolated single oxygen vacancy in 2 x 2 x 3 TiO2 supercell. The

blue ball represents Ti, and the red ball represents O………………………...…..…….33

Fig. 3-2: Density of states of 2 x 2 x 3 supercell TiO2 with oxygen vacancy calculated by (a)

LDA+Ud and (b) LDA+U

d+U

p. U

d and U

p ranges from 3 to 9 eV and U

d is fixed to 8 eV

in the case of (b). Spin-down density of states is not shown………………..………….34

Fig. 3-3: Partial density of state of Ti 3d and O 2p orbitals in 2 x 2 x 3 supercell of TiO2

calculated by LDA+Ud+U

p (U

d = 8 eV, U

p = 6 eV)………………………..…………...35

Fig. 3-4: Partial density of state of two equatorial and one apical Ti atoms surrounding oxygen

vacancy calculated by LDA+Ud (U

d = 7 eV and 8 eV). (a) Equatorial Ti (U

d = 7 eV), (b)

Equatorial Ti (Ud = 8 eV), (c) Apical Ti (U

d = 7 eV), and (d) Apical Ti (U

d = 8 eV).

Additional defect states is observed for Ud = 8 eV………………………….…….……36

Fig. 3-5: Electron localization function and corresponding structural relaxation around neutral

oxygen vacancy calculated by (a) LDA+Ud (U

d = 7 eV) and (b) LDA+U

d (U

d = 8 eV).

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Electrons are depleted in the blue region (background, ELF~0) and electrons are highly

localized in the red region (core region of each atom, ELF~1)………...………...…….38

Fig. 3-6: Partial density of states of Ti atoms surrounding oxygen vacancy calculated by

LDA+Ud+U

p (U

d = 8 eV and U

p = 6 eV) (a) equatorial Ti and (b) apical Ti…………..39

Fig. 3-7: Spin-polarized total density of states of rutile TiO2 2 x 2 x 3 supercell with isolated

single oxygen vacancy obtained by LDA+Ud+U

p (U

d = 8 eV and U

p = 6 eV). Fermi level

is set to 0 eV on the energy scale…………………………………………………...…..40

Fig. 3-8: (a) Schematics of rutile TiO2 (110) plane. The partial density of states of (b) and (c)

nearest neighboring Ti atoms, (d) Ti atom further from oxygen vacancy, (e) and (f)

nearest neighboring O atoms……………………………………………………………42

Fig. 3-9: Charge density difference on rutile TiO2 (110) plane between the relaxed and unrelaxed

supercell when an oxygen vacancy is introduced. Δρ = ρ(relaxed) – ρ(unrelaxed)…….44

Fig. 3-10: Band decomposed partial charge density distribution of TiO2 (110) plane with isolated

single oxygen vacancy………………...…………………………………………...….45

Fig. 3-11: Total density of states of rutile TiO2 with (a) Vo1+

, and (b) Vo2+

calculated by

LDA+Ud+U

p (U

d = 8 eV, U

p = 6 eV)……………………………….…………………49

Fig. 3-12: Electron localization function and structural relaxations around charged oxygen

vacancy in LDA+Ud+U

p (U

d = 8 eV, U

p = 6 eV). (a) Vo

1+, and (b) Vo

2+……..………50

Fig. 3-13: Oxygen vacancy formation energy in rutile TiO2 as a function of Fermi level. (a)

unrelaxed in Ti-rich condition, (b) relaxed in Ti-rich condition, and (c) relaxed in O-

rich condition…………………………………………………………………….……52

Fig. 4-1: Schematic configurations of eight different di-vacancy structures. (a), (b), (c), (d). In-

plane di-vacancy, and (e), (f), (g), (h). Out-of-plane di-vacancy. The inter-vacancy

distance is shown………………………….………………………………………...….58

Fig. 4-2: Interaction energy between two oxygen vacancies as a function of inter-vacancy

distance. (a), (b), (c), and (d) represent in-plane di-vacancy and (e), (f), (g), and (h)

represent out-of-plane di-vacancy………………………………………………………59

Fig. 4-3: Total density of states of in-plane di-vacancy structures……………………………….61

Fig. 4-4: Total density of states of out-of-plane di-vacancy structures…………………………..62

Fig. 4-5: Electron localization function of (a) isolated single oxygen vacancy, and (b) equatorial

di-vacancy………………………………………………………………………...…….63

Fig. 4-6: Schematic pictures of TiO2 octahedral structure showing the direction of Ti 3d orbitals

(a) eg orbitals, and (b) t2g orbitals………………………………………………….……64

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Fig. 4-7: Charge density map of TiO2 and LiTiO2. (a) Difference electron density map

constructed from all valence densities of TiO2, and (b) Difference electron density map

constructed of Li electron density in LiTiO2. [7]……………………………………….65

Fig. 4-8: Iso-surface of band decomposed partial charge density of di-vacancy in TiO2. Iso-

surface is drawn with an iso-value of 0.1e/Å3………………………………………..…67

Fig. 4-9: Schematic diagram of orbital occupancy of charges in four defect states in di-vacancy

structure…………………………………………………………………………………68

Fig. 4-10: Band decomposed partial charge density distribution of di-vacancy in TiO2 (110)…..69

Fig. 4-11: Partial density of states of (a) equatorial Ti, and (b) apical Ti……………………..….70

Fig. 5-1: Schematic configurations of increasing number of oxygen vacancies along two

directions in TiO2 (110). (a) Vacancy ordering in [110], and (b) vacancy ordering in

[001]…………………………………………………………………………………….75

Fig. 5-2: Electron localization function of structures shown in Fig. 5-1. The number of oxygen

vacancies is increased along two directions. (a) Vacancy ordering in [110], and (b)

vacancy ordering in [001]………………….……………………………………...……76

Fig. 5-3: Total density of states of two vacancy-ordered structures. (a) Vacancy-ordered structure

along [110]. (b) vacancy-ordered structure along [001]…….………………………….77

Fig. 5-4: Partial density of states of equatorial and apical Ti atoms surrounding oxygen vacancies.

(a) Vacancy-ordered structure along [001], and (b) vacancy-ordered structure along

[110]………………………………………………………………………………….....79

Fig. 5-5: Schematic illustration of atomic volume calculated by Bader analysis……….………80

Fig. 5-6: Iso-surface of partial charge density distribution of two vacancy-ordered structures. The

picture is drawn at 0.1e/A3 of iso-value. (a) Vacancy-ordered structure along [110], and

(b) vacancy-ordered structure along [001]……...………………………………...…….81

Fig. 5-7: Vacancy formation energies with increasing number of oxygen vacancies along [110]

and [001] directions……………………………………………………………………..82

Fig. 5-8: Vacancy formation energy of TiO2 supercell with different oxygen vacancy

configurations. Vacancies are randomly distributed in R1. In R2 and R3, all vacancies

are confined to the same (110) plane. F represents the vacancy-ordered structure in [001]

direction………………………………………………………………………..………..83

Fig. 5-9: Energy band diagram of TiO2 supercell with an isolated single oxygen vacancy…..….85

Fig. 5-10: Energy band diagram of TiO2 supercell with a di-vacancy…………………………...85

Fig. 5-11: Energy band diagram of TiO2 supercell with ordered vacancies along (a) [110]

direction, and (b) [001] direction……………………………………………………...86

xv

Fig. 5-12: Schematic diagram of the high-symmetry directions in first brillouin zone of the

tetragonal structure………………………………………………………………….…87

Fig. 5-13: Phase diagram of the Ti-O system. TinO2n-1 is the so-called Magnéli phases…..……..88

Fig. 5-14: Iso-surface of partial charge density distribution of a Ti4O7 unitcell. The blue and red

ball represents Ti and O respectively. Excess charges are found in every Ti atom...…89

Fig. 5-15: Schematic illustration of the crystal structure of the Magnéli phase (Ti4O7). The blue

and red ball represents Ti and O respectively, and the yellow ball represents Ti3+

. Two

rows of oxygen atoms are missing in the bipolaron chain which is in the dotted black

circle…………………………………………………………………………………...90

Fig. 6-1: Schematic diagrams of (a) typical unipolar resistive switching behavior, and (b)

electroforming, set and reset switching process. Initial state (as-prepared sample), and

(1) forming, (2) reset, and (3) set processes…………………………………………….95

Fig. 6-2: Schematic diagram of vacancy-ordered structure containing one oxygen atom in the

channel. One oxygen vacancy is exchanged with an oxygen atom adjacent to the

channel………………………………………………………………………………….96

Fig. 6-3: Energy band diagram of (a) vacancy-ordered structure, and (b) vacancy-ordered

structure in which one oxygen vacancy diffuses out of the channel…………..……….97

Fig. 6-4: Schematic diagram of the vacancy-ordered structure containing two oxygen atoms in the

channel……………………………………………………………………….…………98

Fig. 6-5: Energy band diagram of the vacancy-ordered structure in which two oxygen vacancies

diffuse out of the channel…………………………………………………….…………99

Fig. 6-6: Iso-surface of partial charge density of the vacancy-ordered structure in which two

oxygen vacancies diffuse out of the channel…………………………………..………..99

Fig. 6-7: Schematic diagram of one interstitial hydrogen in the perfect TiO2 supercell………..101

Fig. 6-8: Total density of states of TiO2 with one interstitial hydrogen atom…………….…….102

Fig. 6-9: Schematics of TiO2 supercell with one oxygen vacancy and one hydrogen atom.

(a) Before relaxation, and (b) after relaxation.……………………………………102

Fig. 6-10: Total density of states of TiO2 with hydrogen-vacancy complex.……..………103

Fig. 6-11: (a) Schematics of the vacancy chain structure and hydrogen-vacancy complex chain

structure. (b) Total density of states……………………………………………...……105

Fig. 6-12: (a) Electron localization function of hydrogen-vacancy complex chain. (b) Iso-surface

of partial charge density distribution of hydrogen-vacancy complex chain………..…106

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Fig. 6-13: Schematic illustration of resistance switching model. (a) Initial as-sampled state.

Oxygen vacancies are randomly distributed. (b) “ON”-state (Low Resistance State), and

(c) “OFF”-state (High Resistance State)…………………………………...……...…..107

1

CHAPTER 1: INTRODUCTION

Resistive Random Access Memory (ReRAM) is one of the most promising candidates as

a next generation non-volatile memory device that are widely used in our daily lives,

such as computers, digital cameras, cell phones, and storages. ReRAM has gained

significant interest in the past decade due to its potential for high density, low operating

power, fast switching speed, and compatibility with conventional CMOS process.

However, in-depth understanding of the underlying switching mechanism is still lacking.

In this chapter, the background of this research, including motivation and switching

models that have been proposed, is described.

2

1.1 MOTIVATION

Recently, the interest in alternative non-volatile memory (NVM) technologies has

been significantly increased due to the scaling limit of flash memory that is currently the

most popular device in the non-volatile memory market. As the market of state-of-the-art

electronic devices is growing fast, the demand for developing faster and denser non-

volatile memory devices continues to increase rapidly. Different types of non-volatile

memory devices such as phase change RAM (PCRAM), magnetoresistive RAM,

(MRAM), ferroelectric RAM (FeRAM), and resistive RAM (ReRAM) have been actively

studied in a last couple of decades. Among these devices, ReRAM has a strong potential

to be the front-runner due to its lower programming currents and faster switching time.

The structure is simply composed of a resistance changeable material sandwiched

by two terminal electrodes. Resistance switching can be achieved by current or voltage

pulse applied to the electrodes. The resistance state remains stable without being

refreshed. Up to date, a number of different switching characteristics have been observed

in a variety of material systems that includes TiO2 [1], NiO [2], Al2O3 [3], Nb2O5 [4],

SrTiO3 [5], Pr0.7Ca0.3MnO3 [6], Cu2S [7], ZnCdS [8], Ag2S [9], and AgGeSe [10]. In fact,

it has become well understood that a number of combinations of oxides with metal

electrodes can exhibit some kind of resistance switching behavior.

So far several switching models have been proposed; i.e. charge trapping model

[11], conductive filament model [12], Schottky barrier model [13], and electrochemical

migration of point defects [14]. However, none of these models can explain the switching

phenomenon completely due to the lack of fundamental understanding of the undergoing

resistance switching process. In order to elucidate the operating principles accurately, in-

3

depth understanding of the resistance switching mechanism at the atomistic level is

necessary; i.e. how the conducting paths are formed and disconnected. Unfortunately,

however, understanding the resistance switching phenomenon at microscopic level is not

likely to be obtained purely from the experimental method. In this dissertation, a

theoretical approach that involves first principles calculations based on density functional

theory (DFT) has been employed. It is expected that this theoretical work will create a

shortcut to the successful comprehension of resistance switching mechanism, and

accelerate the development of future non-volatile memory devices.

1.2 BASIC OPERATIONAL PRINCIPLES OF RESISTANCE SWITCHING

Resistance switching exhibits a hysteretic current-voltage characteristic with a

sudden change in resistance between a low resistance state (LRS) to a high resistance

state (HRS). This resistance switching can be controlled by either current or voltage.

During the switching process, resistance switches from LRS to HRS. This is called “reset

process”, and resistance switches to the opposite way, from HRS to LRS, which is called

“set process”.

The operation of resistance switching is distinguished by two different schemes

depending on the electrical polarity, as shown in Fig. 1-1 [15]. When both the set and

reset switching take place at the same polarity of the voltage or current, resistance

switching is referred as unipolar switching. That is, switching process is not dependent on

the voltage polarity. Therefore the amount of the applied voltage should be precisely

controlled in the unipolar switching for proper operation. In this switching, high

resistance state switches into low resistance state by a threshold voltage with a current

compliance required to protect the sample from breaking down. Once the system turns

4

into low resistance state, reset switching from low resistance state to high resistance state

occurs at a lower voltage than set switching.

In contrast, bipolar switching shows only one switching process either ser or reset

at one voltage polarity. For instance, if set switching occurs at a positive voltage polarity,

reset switching is obtained at a negative voltage polarity. In both switching modes, two

resistance states are distinguished to each other at a small read-out voltage, therefore read

operation has no influence on the resistance state. It should be noted that it requires

different switching mechanisms to be able to explain different switching modes and the

switching curve could depend on materials and measurement methods.

1.3 CLASSIFICATION OF THE RESISTANCE SWITCHING MECHANISM

One of the most challenging issues in this area is that resistance switching

mechanism has not been clarified yet. Up to now, several models have been proposed.

Fig. 1-2 represents five different switching mechanisms suggested in ITRS roadmap 2010.

(a) (b)

Fig. 1-1 Schematic diagram of two different resistance switching modes. (a) unipolar, and (b)

bipolar. (based on Ref. [15])

5

The first one is phase change mechanism. The resistance is determined by the

transition between the crystalline and amorphous phase of switching materials that is

controlled by a thermal process. GeSbTe (GST) is the most representative phase change

material used in phase change RAM (PCRAM).

The second is thermo-chemical mechanism which is known as fuse-antifuse type.

NiO falls into this mechanism. The applied voltage induces partial breakdown of the

switching material, which makes the system low resistance state. Then, the filament type

conductive path is disrupted by joule heating that is caused by high current density

through conductive filament. Filamentary conduction in “ON”- state has been reported in

NiO [12].

The third one is valency change mechanism. This mechanism is referred to anion-

migration induced redox type. In this mechanism, anions, typically oxygen vacancies in

many transition metal oxides, drift to and pile up from the cathode, therefore oxygen

deficient region starts to grow and to expand to the anode. Oxygen vacancies reduce the

Fig. 1-2 Classification of resistive memory technologies according to their primary principle

of operation (based on ITRS 2010 updates)

6

valence state of anions turning oxide into metallic phase by forming metallic conductive

channel. Thus the conduction path is composed of highly oxygen deficient non-

stoichiometric phase which shows higher conductivity than normal phase. Janousch et al.

have observed a high oxygen vacancy concentration region between anode and cathode

as shown in Fig. 1-3, which was obtained from X-ray absorption near edge spectroscopy

(XANES) spectra [16]. This shows that the behavior of oxygen vacancies in transition

metal oxide plays an important role in resistance switching. It is known that TiO2, SrTiO3,

and Pr0.7Ca0.3MnO3 belong to this category.

Electro-chemical metallization is referred to cation-migration induced redox

model. During electroforming process, the oxidation of metal electrode occurs at the

anode interface, and then positively charged metal cations migrate to the cathode and are

reduced at the interface. The reduced metal atoms grow toward the anode and form a

metallic conductive path, which turn the system into “ON”-state. Then, the reversed bias

Fig. 1-3 X-ray fluorescence mapping after electroforming in Cr-doped SrTiO3 showing the

distribution of oxygen vacancies between two electrodes. [16]

7

dissolves these metal atoms at the interface resulting in “OFF”-state. Ag or Cu is

typically used as the electrochemically active anode with GeSex as a switching material.

The last one is electrostatic/electronic mechanism. Electronic charges are injected

and trapped at interface defect sites. This affects Schottky barrier and changes resistance.

Some perovskite materials can be explained by this mechanism [17].

Although significant progress has been recently made in understanding switching

mechanism, the boundary between these mechanisms is obscure and in-depth

understanding is still lacking. However, the microscopic mechanism of the resistance

switching must be well understood in order to make ReRAM practical and useful, thus

much effort is still necessary to explore the switching mechanism.

1.4 SWITCHING MATERIALS AND TITANIUM DIOXIDE

A variety of metal-insulator-metal systems have shown electrically induced

resistance switching characteristics and there have been a large number of experimental

Table 1-1 Summary of resistance switching materials

8

studies on the development of switching materials or characterization of switching

properties. A number of materials that exhibit the resistance switching effect are listed in

Table 1-1. Since many materials have shown different resistance switching behaviors,

there is not much value to further expand the list of these materials. Instead, it is quite

necessary to determine a few promising candidates, and then focus on the improvement

of the performance and understanding of the switching mechanism.

Among those many switching materials, TiO2 has been receiving a great deal of

attention over the last few decades. TiO2 is a semiconductor oxide with a wide range of

application such as photocatalyst, gas sensor, and semiconducting electrode in solar cell

devices. It has been well known that many electrical and chemical properties of most of

the transition metal oxides are affected by point defects [18]. In fact, several previous

studies have shown that TiO2 exhibits n-type semiconducting property with extra electron

carriers generated by the formation of oxygen vacancies [19, 20]. Recently, an increased

Fig. 1-4 (a) High resolution TEM image of a nanofilament. (b) Local I-V curves measured on

nanofilament and TiO2 in background.

9

interest in the properties of TiO2 emerged with the recognition of its remarkable

“memristive” switching behavior [21]. It was claimed that electrical switching occurs by

means of the drift of positively charged oxygen vacancies forming and dispersing a

conductive channel.

Another recent work reported by Kwon et al. has shown the formation of a

conical shaped conductive filament with a diameter of 5~10nm in TiO2 observed by high

resolution TEM as shown in Fig. 1-4. They confirmed that the formed nanofilament is a

Magnéli phase referred as Ti4O7 in which oxygen atoms are highly deficient [22]. They

also measured in-situ I-V characteristics using conductive atomic force microscopy (C-

AFM) showing much lower resistance in the nanofilament. Therefore it is quite evident

that oxygen vacancies in TiO2 play a crucial role in the resistance switching. The above

references suggest that TiO2 has a great potential to be a strong candidate for the next

generation of non-volatile memories. In this regard, rutile TiO2 was chosen as the

resistance switching material in this theoretical study.

1.5 OUTLINE OF DISSERTATION

This dissertation consists of the theoretical investigation of “ON”-state

conduction mechanism and the switching between “ON” and “OFF”-state in TiO2 by

means of exploring oxygen vacancy behavior. Prior to this main work, optimization of

first principles calculation method was performed to improve the accuracy of simulation

results. Chapter 2 introduces the overall computation methodology used in this study. In

order to achieve a better description of the physical properties and energy band gap of

TiO2, U parameters which correct for the on-site Coulomb interaction between electrons

10

in d orbitals are used in conjunction with the local density approximation (LDA) method,

and various U values are explored to find the optimum value. Chapter 3 and 4 describes

the effect of isolated single oxygen vacancy and di-vacancy in TiO2. Density of states,

electron localization function, and band decomposed charge density distribution are

presented to show how oxygen vacancies change the electronic structure and how they

affect the conduction property of TiO2. The effect of multiple oxygen vacancies is

discussed in Chapter 5. Particularly the effect of oxygen vacancy ordering on the

conductivity is described in detail. We find that the resistance of TiO2 is significantly

decreased by a vacancy ordering along a certain direction. The resistance switching from

“ON”-state to “OFF”-state is discussed in Chapter 6. The energy band structure and

charge density distribution show that not only the diffusion of oxygen vacancies but also

the interaction of hydrogen with oxygen vacancy induces the transition from “ON”-state

to “OFF”-state. Chapter 7 concludes the dissertation with a summary of work and future

plan for further research progress.

11

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no. 11, pp. 113501, Mar 2007.

[2] S. Seo, M. J. Lee, D. H. Seo, E. J. Jeoung, D.-S. Suh, Y. S. Joung, I. K. Yoo, I. R.

Hwang, S. H. Kim, I. S. Byun, J.-S. Kim, J. S. Choi, and B. H. Park, “Reproducible

resistance switching in polycrystalline NiO films,” Appl. Phys. Lett., vol. 85, no. 23,

pp. 5655-5657, Dec 2004.

[3] T. W. Hickmott, “Low-frequency negative resistance in thin anodic oxide films,” J.

Appl. Phys., vol. 33, pp. 2669, Sep 1962.

[4] H. Sim, D. Choi, D. Lee, S. Seon, M.-J. Lee, I.-K. Yoo, and H. Hwang, “Resistance-

switching characteristics of polycrystalline Nb2O5 for Nonvolatile memory

application,” IEEE Electron Device Lett., vol. 26, no. 5, pp. 292-294, May 2005.

[5] Y. Watanabe, J. G. Bednorz, A. Bietsch, C. Gerber, D. Widmer, A. Beck, and S. J.

Wind, “Current-driven insulator-conductor transition and nonvolatile memory in

chromium-doped SrTiO3 single crystals,” Appl. Phys. Lett., vol. 78, no. 23, pp. 3738-

3740, Jun 2001.

[6] A. Baikalov, Y. Q. Wang, B. Shen, B. Lorenz, S. Tsui, Y. Y. Sun, Y. Y. Xue, and C.

W. Chu, Appl. Phys. Lett., vol. 83, no. 5, pp. 957-959, Aug 2003.

[7] T. Sakamoto, H. Sunamura. H. Kawaura, T. Hasegawa, T. Nakayama, and M. Aono,

“Nanometer-scale switches using copper sulfide,” Appl. Phys. Lett., vol. 82, no. 18,

pp. 3032-3034, May 2003.

12

[8] Z. Wang, P. B. Griffin, J. McVittie, S. Wong, P. C. McIntyre, and Y. Nishi,

“Resistive switching mechanism in ZnxCd1-xS nonvolatile memory devices,” IEEE

Electron Device Lett., vol. 28, no. 1, pp. 14-16, Jan 2007.

[9] K. Terabe, T. Hasegawa, T. Nakayama, and M. Aono, “Quantized conductance

atomic switch,” Nature, vol. 433, no. 7021, pp. 47-50, Jan 2005.

[10] M. N. Kozicki, C. Gopalan, M. Balakrishnan, M. Park, and M. Mitkova, “Non-

volatile memory based on solid electrolytes,” NVMTS, pp. 10-17, 2004.

[11] A. Chen, S. Haddad, Y. C. Wu, Z. Lan, T. N. Fang, and S. Kaza, “Switching

characteristics of Cu2O metal-insulator-metal resistive memory,” Appl. Phys. Lett.,

vol. 91, pp. 123517, Sep 2007.

[12] D. C. Kim, S. Seo, S. E. Ahn, D. S. Suh, M. J. Lee, B. H. Park, I. K. Yoo, I. G. Baek,

H. J. Kim, E. K. Yim, J. E. Lee, S. O. Park, H. S. Kim, U.-In. Chung, J. T. Moon,

and B. I. Ryu, “Electrical observations of filamentary conductions for the resistive

memory switching in NiO films,” Appl. Phys. Lett., vol. 88, pp. 202102, May 2006.

[13] T. Fujii, M. Kawasaki, A. Sawa, H. Akoh, Y. Kawazoe, and Y. Tokura, “Hysteretic

current-voltage characteristics and resistance switching at an epitaxial oxide

Schottky junction SrRuO3/SrTi0.99Nb0.01O3,” Appl. Phys. Lett,. vol. 86, pp. 012107,

Dec 2004.

[14] X. Guo, C. Schindler, S. Menzel, and R. Waser, “Understanding the switching-off

mechanism in Ag+ migration based resistively switching model systems,” Appl.

Phys. Lett., vol. 91, pp. 133513, Sep 2007.

[15] R. Waser, and M. Aono, “Nanoionics-based resistive switching memories,” Nat.

Mat., vol. 6, pp. 833-840, Nov 2007

13

[16] M. Janousch, G. I. Meijer, U. Staub, B. Delley, S. F. Kang, and B. P. Andreasson,

“Role of oxygen vacancies in Cr-doped SrTiO3 for resistance-change memory,” Adv.

Mater. Vol. 19, pp. 2232-2235, Sep 2007

[17] A. Sawa, T. Fujii, M. Kawasaki, and Y. Tokura, “Interface resistance switching at a

few nanometer thick perovskite manganite active layers,” Appl. Phys. Lett., vol. 88,

pp. 232112, Jun 2006

[18] N. Yu, and J. W. Halley, “Electronic structure of point defects in rutile TiO2,” Phys.

Rev. B, vol. 51, pp. 4768-4776, Feb 1995

[19] M. D. Earle, “The electrical conductivity of titanium dioxide,” Phys. Rev., vol. 61,

pp. 56-62, Jan 1942

[20] D. C. Cronemeyer, “Infrared absorption of reduced rutile TiO2 single crystals,” Phys.

Rev., vol. 113, pp. 1222-1226, Mar 1959

[21] J. J. Yang, M. D. Pickett, X. Li, D. A. A. Ohlberg, D. R. Stewart, and R. S. Williams,

“Memristive switching mechanism for metal/oxide/metal nanodevices,” Nat.

Nanotech., vol. 3, pp. 429-433, Jul 2008

[22] D.-H. Kwon, K. M. Kim, J. H. Jang, J. M. Jeon, M. H. Lee, G. H. Kim, X.-S. Li, G.-

S. Park, B. Lee, S. Han, M. Kim, and C. S. Hwang, “Atomic structure of conducting

nanofilaments in TiO2 resistive switching memory,” Nature Nanotech., vol. 5, pp.

148-153, Jan 2010

14

CHAPTER 2: COMPUTATIONAL METHODOLOGY

The computational methods used in this dissertation are discussed in this chapter. The

limitation of the conventional local density of approximation (LDA) method, one of the

most popular methods to calculate the exchange and correlation energy of electrons, is

described. In order to overcome this limitation, on-site Coulomb correction term is

included and the LDA+U method is used. The effect of the Coulomb correction on both

defect properties and energy band gap of TiO2 is discussed.

2.1 COMPUTATIONAL DETAILS

Density functional calculations using the Vienna ab initio simulation package,

(VASP) [1-3] were carried out for the study of oxygen deficient rutile TiO2. It derives

properties of the molecule based on a determination of the electron density of the

15

molecule. An energy cutoff of 353 eV was employed for the plane wave expansion of

electron wave functions. The ionic potentials are described by projector augmented-wave

pseudopotentials (PAW) [2]. For k-point integration within the first Brillouin zone, 8 x 8

x 12 Monkhorst-Pack grid for primitive cell and 4 x 4 x 4 Monkhorst-Pack grid for

supercell were selected. A pseudo atomic calculation was performed for O 2s22p

4 and Ti

3s23p

63d

24s

2 states of valence electrons. All atoms were allowed to relax with energy

convergence tolerance of 10-6

eV/atom and ground state was obtained by minimizing the

force on each atom less than 0.001 eV/Å .

The oxygen vacancy was introduced within 3 x 3 x 4 supercell of rutile TiO2 so

that it consists of 72 Ti and 144 O atoms. The exchange and correlation energies of

electrons are described in the local density approximation with the correction of on-site

Coulomb interactions (i.e., LDA+U) between electrons. On-site Coulomb correction

between the p-orbital electrons of O (Up) has been applied in addition to the correction

for the d-orbital electrons in Ti (Ud), which is referred as LDA+U

d+U

p. In order to find

the optimum Ud and U

p parameters, we applied various U

d and U

p ranging from 3 eV to 9

eV and investigated their effect on the band structure. For the exchange parameter J, a

value of 0.6 eV was used [4].

2.2 LIMITATIONS OF LOCAL DENSITY OF APPROXIMATION (LDA)

2.2.1 UNDERESTIMATION OF ENERGY BAND GAP

The most commonly used and successful approximation for the calculation of

exchange and correlation energies of electrons is the local density approximation (LDA),

first formulated by Kohn and Sham [5] in 1965. They showed that the total energy

16

functional for the interacting many-particle system can be transformed into a set of one-

particle equations, called the Kohn-Sham equations.

iiieff rVm

)(

2

22

(2-1)

The Kohn-Sham equations are solved self-consistently and the ground state electron

density is determined by the one-electron wave functions, i , as

i

irrrn2

)()()( (2-2)

The LDA states that the exchange-correlation energy at certain point is

considered the same as that for a locally homogeneous electron gas of the same charge

density. Although this approximation is extremely simple, it works well for slowly

varying electron densities in bulk material. However, when the electron density varies

rapidly, for instance as in molecules, or for strongly correlated electron systems as ionic

semiconductors, it fails to describe atomic ground state energies, ionization energies, and

binding energies accurately. A well known issue is the underestimation of the band gap in

semiconductors which is troublesome for energy band structure calculations in TiO2.

In the past decade several theoretical investigations employed the local density

approximation to calculate the electronic structure of TiO2 [6-8]. Going beyond LDA,

recent theoretical developments of exchange and correlation functionals, as the addition

to the on-site Coulomb correction within LDA+U [9], dynamical mean field theory

approaches [10], the progress in hybrid functional [11], and GW implementations [12]

have been shown to correct for some of the most severe shortcomings of the conventional

LDA, i.e. the accurate prediction of the energy band gap and electronic defect states in

transition metal oxides.

17

Previous LDA+U studies [13-14] , however, did point to the fact that the energy

band gap of rutile TiO2 strongly depends on the U parameter, when the correction is

applied on the Ti d-orbitals, but remains still underestimated compared to the

experimental energy band gap of 3.0 eV, even at high values of U. A few earlier

theoretical studies on transition metal systems [15-16] discussed the effect of on-site

Coulomb corrections on the p-electrons (Up) of the oxygen in addition to the d-electrons

(Ud) of the transition metal. According to Nekrasov et al., [17] when self interaction

correction is implemented only to the d orbitals, the oxygen p orbitals do not shift from

the LDA obtained positions and the occupied d band lies much lower than oxygen

valence band. In order to improve this situation, self interaction correction must be

applied to all valence states including oxygen p orbitals. Therefore the correct description

of the energy splitting between occupied and unoccupied orbitals can be obtained.

Several theoretical studies have already reported about the effect of Up values [18-20].

2.2.2 WRONG DESCRIPTION OF DEFECT STATES IN THE ENERGY BAND GAP

In addition, the position of defect states induced by oxygen vacancies in TiO2

must be carefully examined. Reduced TiO2, in the phase diagram of the Ti-O system [21],

corresponds to several stable titanium-oxygen phases that exist between Ti2O3 and TiO2,

therefore the oxygen vacancy can become the major point defect observed in TiO2. The

role of oxygen vacancies in TiO2 has been extensively studied during the past decades by

various experimental and theoretical methods; however the effect of oxygen vacancy on

the electronic band structure remains controversial. The experimental results of

Cronemeyer [22] showed that two energy levels are generated by oxygen vacancy at 0.75

eV and 1.18 eV below the conduction band minimum. However, Ghosh et al. [23]

18

observed eight different energy levels within the band gap on the oxygen deficient rutile

TiO2 sample. Theoretical study of Chen et al. [24] on reduced rutile TiO2 showed that a

defect state was produced at 0.87 eV below the conduction band minimum for neutral

oxygen vacancy, and at 1.78 eV below the conduction band minimum for positively

charged oxygen vacancy. They employed an embedded-cluster numerical discrete

variation method. Another investigation based on a semi-empirical self-consistent method

using the tight-binding model observed the oxygen vacancy defect state at 0.7 eV below

the conduction band minimum. On the other hand, LDA calculation of Cho et al. [25]

observed the oxygen vacancy defect state above the conduction band. Another theoretical

study of Ramamorthy et al. [26] using ab initio self-consistent pseudopotential total-

energy calculations found that oxygen vacancies introduce defect state at 0.3 eV below

the conduction band minimum.

In this dissertation, we discuss the effect of electronic correlations on the energy

band gap, atomic relaxation, charge density, and vacancy formation energy of rutile TiO2

by using density functional theory (DFT) and the local density approximation (LDA)

with the addition of on-site Coulomb corrections (LDA+U) method.

2.3 EFFECT OF ON-SITE COULOMB CORRECTION (U)

2.3.1 ELECTRONIC STRUCTURE OF RUTILE TIO2

We first preformed ab initio calculations for bulk rutile TiO2. The unit cell

structure of rutile TiO2 is illustrated in Fig. 2-1. Six O atoms, forming a distorted

octahedral structure, surround the Ti atoms. Due to the different Ti-O bond lengths, Ti 3d

orbitals will split into two sets of t2g and eg orbitals. eg orbitals are associated with the

19

upper conduction band and point at surrounding six oxygen atoms forming ζ bonds. On

the other hand, oxygen atoms are surrounded by three Ti atoms in a planar geometry. 2p

orbitals of O atoms make three ζ bonds with eg orbitals of the surrounding Ti atoms.

The partial density of Ti and O states of rutile TiO2 for LDA are shown in Fig. 2-2.

The Fermi level position is set to 0 eV. The valence band states are mainly composed of

O 2p hybridized with Ti 3d orbitals, and the top of the valence band is dominated by O

2p states. The valence band width of 6 eV is in good agreement with experimental

measurements, which are in the range of 5-6 eV [27-28]. On the other hand, the

conduction band states have Ti 3d states with weak hybridization with O 2p states. The

two peaks in the conduction band are a result of the ligand field splitting of the Ti 3d

orbitals into two sets of t2g and eg states. We confirmed that there is no significant

difference in the electronic structure obtained with LDA or GGA methods, both

underestimate the band gap by about 44 %.

Fig. 2-1 Unit cell structure of rutile TiO2. Small light blue ball denotes a Ti atom and big red

one an O atom. Each Ti atom is surrounded by six O atoms.

20

2.3.2 U VALUE FROM PREVIOUS LITERATURE

The underestimated energy band gap is mainly due to inadequate description of

the strong Coulomb interaction between 3d electrons localized on metal ions, which is

characterized by the Hubbard U. The strength of this interaction is known to be

comparable with the valence bandwidth [29], therefore the process associated with

electron transfer between two metal ions or addition or removal of d electrons gives rise

to large fluctuations of the energy of the system, leading to the localization of carriers and

to the formation of band gaps [30]. According to ref. [9], LDA+U functional can be

written as

)(2

)( 2

,, mmLDAULDA nnJU

EE (2-3)

-6 -4 -2 0 2 4 6 80

2

4

6

eg

Energy (eV)

DO

S (

arb

. u

nit

s)

LDA_Ti 3d

LDA_O 2pt2g

Fig. 2-2 Partial density of states of Ti 3d orbital (solid line) and O 2p orbital (dotted line)

calculated with LDA. The top of the valence band is set to 0 eV on the energy scale.

21

where m is the orbital momentum and U and J are the spherically averaged matrix

elements of the screened Coulomb electron-electron interaction. nζ is the expectation

value of the operator for the number of electrons occupying particular site.

There is no general consensus for LDA+U about how to find the optimum U

value. C. J. Calzado et al. [13] suggested that a Ueffd value (Ueff

d=U

d-J) of 5.5±0.5 eV

gives a correct description of the defect state using the LDA+U implementation in the

VASP code. On the other hand, N. A. Deskins et al. [31] chose a Ueffd value of 10 eV

using the same code which finds a band gap in satisfactory agreement with experiment

for rutile TiO2. In most LDA+Ud studies of transition metal oxides, U

d values are fitted in

an empirical way, but schemes based on theoretical determination of the U parameter had

been also performed [32].

2.3.3 EFFECT OF Ud IN LDA+U

In order to investigate the effect of Coulomb correction for Ti 3d orbital electrons,

we have employed several calculations, in which a range of Ud values were chosen. For

validation, we have considered the structural parameters of rutile TiO2, i.e. lattice

constants and bulk modulus, the band gap energy, the position of the defect state and

vacancy formation energy. All these parameters except for bulk modulus can be directly

calculated from the total energy calculation. Bulk modulus can be obtained from the

equation of state for energy [38].

11

1

)/)(

'

0

00

'

0

0

'

0

00

'0

B

VB

B

VV

B

VBEVE

B

(2-4)

sV

EP

TV

PVB

(2-5)

22

where V0 is equilibrium volume and E0 is the total energy at equilibrium lattice constant.

B0 is bulk modulus when pressure is zero. A solid has a certain equilibrium volume V0,

and the energy increases as volume is increased or decreased by small amount from that

value. In this manner, we can plot energy with respect to volume, and the bulk modulus

can be obtained from the second derivative of this E-V curve.

Table 2-1 Summary of structural parameters a(Å ), c(Å ), bulk modulus B (GPa), and band gap

energy Eg (eV) calculated by LDA+Ud with experimental results.

Ud (eV) a (Å ) c (Å ) c/a B (GPa) Eg (eV)

Expt. [33] 4.584 2.953 0.644 284 3.00

GW [34] 4.594 2.958 0.644 4.80

HSE[35] 4.590 2.947 0.642 3.05

HSE[36] 4.600 2.962 0.644 256 3.25

PW1PW[37] 4.59 2.980.649

234 3.54

LDA[25] 4.563 2.914 0.637 265 1.7

GGA 4.644 2.975 0.641 222 1.78

LDA 4.557 2.929 0.643 258 1.79

LDA+Ud

3 4.572 2.957 0.647 255 1.96

4 4.579 2.969 0.648 254 2.04

5 4.585 2.982 0.650 253 2.12

6 4.591 2.994 0.652 252 2.21

7 4.597 3.005 0.654 251 2.30

8 4.603 3.016 0.655 250 2.40

9 4.610 3.027 0.657 248 2.50

23

Table 2-1 shows the structural parameters and band gap energies calculated with

increasing Ud value and in comparison with experimental results. Both lattice constants

are found to increase with increasing Ud. However, the dependence of the bulk modulus

on the Ud is negligible. The band gap energy is increased with increasing U

d values from

3 eV to 9 eV, from a value of 1.79 eV obtained with the conventional LDA to 2.50 eV.

This is in agreement with the previous theoretical investigations [26], however, the band

gap energy is still underestimated as compared to the experimental band gap energy of

3.0eV. This indicates that the LDA+Ud approach may not be adequate to calculate the

electronic structure of rutile TiO2.

Fig. 2-3 Density of states obtained by (a) LDA+Ud, and (b) LDA+U

d+U

p. The top of the

valence band is set to 0 eV on the energy scale. Ud and U

p range from 3 to 9 eV and U

d is fixed

to 8 eV in the case of (b).

24

2.3.4 EFFECT OF UP IN LDA+U

To improve on the on-site Coulomb corrections we propose to include additional

corrections for the oxygen 2p orbitals (Up). Fig. 2-3(a) shows the density of states

calculated by LDA+Ud, where U

d ranges from 3 eV to 9 eV. Fig. 2-3(b) corresponds to

LDA+Ud+U

p calculations, where U

d is fixed to 8 eV and U

p ranges from 3 eV to 9 eV.

The Fermi level is scaled to zero of energy. In order to show more clearly the difference

among various U parameters, we show only the positive spin density of states although

the density of states of TiO2 had been calculated as a spin polarized system. The energy

band gap is increased with increasing both Ud and U

p.

In order to obtain a better understanding of the effects of Ud and U

p, we evaluate

the band structure calculated by employing LDA+Ud and LDA+U

d+U

p, and presented in

Fig. 2-4 Energy band structure and density of states calculated by (a) LDA+Ud and (b)

LDA+Ud+U

p. Fermi level is not scaled to 0 eV on the energy scale.

25

Fig. 2-4. Fig. 2-4(a) shows the band structure corresponding to Ud = 3 eV, 6 eV and 9 eV,

and in Fig. 2-4(b) Ud = 8 eV is fixed, while U

p = 3 eV, 6 eV and 9 eV. As U

d is increased,

while the conduction band minimum is shifted upward, the valence band is not affected

as shown in Fig. 2-4(a). However, calculations involving LDA+Ud+U

p show an upward

shift of the conduction band as well as a downward shift of the valence band, as shown in

Fig. 2-4(b). The on-site Coulomb correction Ud+U

p increases the splitting between

occupied and empty energy states, since the correction is dependent on the occupancy of

the orbitals [17].

It has been shown that rutile TiO2 presents characteristics of a charge transfer

insulator [25]. Thus increased energy separation between Ti 3d and O 2p bands results in

higher charge transfer energy and ultimately leads to a more ionic character. The

electronic population function corresponding to each individual atom was calculated

using the Bader charge analysis. We find that by applying both Ud and U

p (U

d = 8 eV, U

p

= 6 eV) the electron population on each Ti atom is reduced by 0.358e while that on each

Table 2-2 Summary of structural parameters a(Å ), c(Å ), bulk modulus B (GPa), and band gap

energy Eg (eV) calculated by LDA+Ud+U

p.

Ud (eV) Up (eV) a (Å ) c (Å ) c/a B (GPa) Eg (eV)

8 3 4.574 3.004 0.657 290 2.58

8 4 4.562 2.997 0.657 282 2.67

8 5 4.547 2.993 0.658 289 2.77

8 6 4.532 2.986 0.659 269 2.87

8 7 4.518 2.980 0.660 269 2.98

8 8 4.500 2.973 0.661 281 3.11

8 9 4.483 2.965 0.661 296 3.24

26

O atom is increased by 0.179e, indicating that the on-site Coulomb correction predicts a

more ionic bonding character.

The calculated band gap is around 3.0 eV for Ud+U

p (8+6, 8+7, 8+8 eV) in good

agreement with the experimental value [33]. The calculated structural parameters and

band gap energies for several combinations of Ud and U

p are shown in Table. 2-2. We

find that the correction on the energy band gap due to Up is comparable to that due to U

d,

indicating that the adequate description of the on-site Coulomb interactions of the p-

orbital electrons of O atom is as significant as is for the metal d-orbital electrons, a

feature also observed by Nekrasov et al. [17] for other transition metal oxides.

27

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32

CHAPTER 3: SINGLE OXYGEN VACANCY

This chapter discusses the effect of an isolated single oxygen vacancy in rutile TiO2. 2 x

2 x 3 supercell of TiO2 that contains 72 atoms is used to incorporate oxygen vacancy.

Before we discuss oxygen vacancy, the effect of Coulomb interaction parameter (U) on

the defect states is presented first. The removal of an oxygen atom leaves two electrons in

the bulk, and then the electronic properties are severely affected by these excess electrons.

In this chapter, the behavior of these excess electrons is investigated by the analysis of

electron localization function, density of state, and charge density distribution.

33

3.1 EFFECT OF U ON DEFECT STATES

3.1.1 POSITION OF DEFECT STATES IN DENSITY OF STATES

It is well known that the oxygen vacancies play an important role in the electrical

conductivity of TiO2. We first employed LDA+Ud method in the calculation of the

electronic structure of 2 x 2 x 3 TiO2 supercell containing one oxygen vacancy. Fig. 3-1

displays the schematic illustration of an isolated single oxygen vacancy in 2 x 2 x 3 TiO2

supercell.

We have examined the effect of the U parameter on the position of the defect

states originating from oxygen vacancy. Total density of states for several values of Ud

ranging from 3 to 9 eV is shown in Fig. 3-2(a). In this figure, the highest occupied energy

level is set to zero. In other words the defect energy level is set to zero. As Ud value

increases, the conduction bands shift toward higher energy states, and thus the defect

state is separated from the conduction band for Ud values larger than 5 eV. Note that the

distance between the localized defect states and the top of the valence band is constant,

Fig. 3-1 Schematic structure of isolated single oxygen vacancy in 2 x 2 x 3 TiO2 supercell.

The blue ball represents Ti, and the red ball represents O.

34

regardless of the Ud value.

With the removal of an oxygen atom a neutral oxygen vacancy is created, and two

electrons that previously occupied oxygen 2p orbitals become localized. These electrons

may be responsible for the n-type transport behavior depending on the position of the

defect state in the band gap. Therefore, the correct prediction of defect state has important

implications for applications, and methods beyond LDA are necessary to be employed.

Pointing to the limitation of using large Ud, an additional defect state in the band gap is

observed for Ud = 8 and 9 eV. We found that this defect state is an artifact caused by the

large Ud, therefore its origin is unphysical and different from that of the previous defect

state.

The effect of the increasing Up is shown in Fig. 3-2(b). In this case, U

d is fixed at

Fig. 3-2 Density of states of 2 x 2 x 3 supercell TiO2 with oxygen vacancy calculated by (a)

LDA+Ud and (b) LDA+U

d+U

p. U

d and U

p ranges from 3 to 9 eV and U

d is fixed to 8 eV in

the case of (b). Spin-down density of states is not shown.

35

8 eV. In contrast to Fig. 3-2(a), the valence band is gradually shifted downward, and the

position of defect states below the conduction band minimum is not affected by Up.

However, the additional defect state observed when Ud is higher than 7 eV in Fig. 3-2(a)

has disappeared. We found that defect states generated by oxygen vacancy shows mainly

Ti 3d character and minor O 2p, indicating a high localization of the two electrons from

the three nearest neighboring Ti atoms surrounding the oxygen vacancy. Fig. 3-3 shows

the density of states of Ti 3d and O 2p orbital.

3.1.2 PARTIAL DENSITY OF STATES OF NEIGHBORING TI

In rutile TiO2, due to different bond lengths, three Ti atoms surrounding the

oxygen vacancy are grouped as two atoms of equatorial Ti and one atom of apical Ti.

These Ti atoms and oxygen vacancy are in the same plane (110). The partial densities of

states of the Ti atoms surrounding the oxygen vacancy, are displayed in Fig. 3-4,

calculated for Ud = 7 and 8 eV. In the case of U

d = 7 eV, there is only one defect state on

both type of Ti atoms, indicating that two electrons are strongly localized on the 3d

orbitals of three Ti atoms. These two electrons are occupied by spin-up and spin-down

-8 -6 -4 -2 0 2 4 60

5

10

15

20

25

30

DO

S (

arb

. u

nit

s)

Energy (eV)

Ti 3d

-8 -6 -4 -2 0 2 4 60

5

10

15

20

25

30

O 2p

DO

S (

arb

. u

nit

s)

Energy (eV)

Fig. 3-3 Partial density of state of Ti 3d and O 2p orbitals in 2 x 2 x 3 supercell of TiO2

calculated by LDA+Ud+U

p (U

d = 8 eV, U

p = 6 eV).

36

defect states which are localized at the same energy level. Then, an additional defect state

observed for Ud is 8 eV on both types of Ti atoms.

In Fig. 3-4(b), one of the defect states is located at the bottom of the conduction.

We confirmed that spin-up and down defect states are observed at the same energy level

up to Ud = 7 eV. Several previous theoretical studies have reported that oxygen vacancy

gives rise to one energy level of defect states in TiO2 [1-3]. For Ud = 8 eV, however, the

spin-down defect state disappears and two spin-up defect states at a different energy level

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Energy (eV)

(a) Equatorial Ti (Ud=7eV)

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Energy (eV)

(b) Equatorial Ti (Ud=8eV)

-8 -6 -4 -2 0 2 4 6 80

1

2

3

4

5(c) Apical Ti (U

d=7eV)

DO

S (

arb

. u

nit

s)

Energy (eV)-8 -6 -4 -2 0 2 4 6 8

0

1

2

3

4

5(d) Apical Ti (U

d=8eV)

DO

S (

arb

. u

nit

s)

Energy (eV)

Fig. 3-4 Partial density of state of two equatorial and one apical Ti atoms surrounding oxygen

vacancy calculated by LDA+Ud (U

d = 7 eV and 8 eV). (a) Equatorial Ti (U

d = 7 eV), (b)

Equatorial Ti (Ud = 8 eV), (c) Apical Ti (U

d = 7 eV), and (d) Apical Ti (U

d = 8 eV). Additional

defect states is observed for Ud = 8 eV.

37

are observed as shown in Fig. 3-4(b) and (d). This is clearly an artifact observed at large

Ud, supported also by the altering character of the partial conduction band structure. Due

to the consideration of extra strong Coulomb interaction, two electrons are likely to be

occupied by different level of orbital with the same spin direction. In addition, in case of

Ud = 8 eV, the shape of conduction band of apical Ti shown in Fig. 3-4(d) is severely

distorted in contrast to other band structures. We also observed similar changes in density

of states for Ud of 9 eV. Therefore, an excess of Coulomb correction on the d orbitals

may cause local distortion in the electronic structure, and therefore its usage should be

well tested.

3.2 ELECTRON LOCALIZATION FUNCTION

3.2.1 ELECTRON LOCALIZATION ON TI

In order to understand the physical effects of how Ud affects the atomic bonding

in TiO2, the valence charge electron localization functions (ELF) were calculated by

using this formula [4]:

121

ELF (3-1)

where

0

D

D (3-2)

and

3/53/220 )6(5

3 D , (3-3)

38

where is the electron spin density and 0

D corresponds to a uniform electron gas with

spin density equal to the local value of )(r . The ratio is thus a dimensionless

localization index calibrated with respect to the uniform density electron gas as reference.

Therefore the localization of an electron is associated with the probability density to find

a same spin electron near the reference point. ELF gives account of the type of bonding

preferred and has a value between 0 and 1. The value of 1 corresponds to perfectly

localized regions, showing mostly covalent bonding character. 0.5 corresponds to

electron gas indicating metallic nature, while values between 0 and 0.5 indicate regions

of low electron density, where strong ionic interactions are dominated.

The ELFs for Ud = 7 and 8 eV are shown in Fig. 3-5. The red region (dark region

in gray scale) in the oxygen vacancy site shows that two electrons are highly localized on

the 3d orbitals in the proximity of the vacancy site. The key difference of ELFs between

LDA+Ud of 7 eV and LDA+U

d of 8 eV is the bonding character for the nearest

Fig. 3-5 Electron localization function and corresponding structural relaxation around neutral

oxygen vacancy calculated by (a) LDA+Ud (U

d = 7 eV) and (b) LDA+U

d (U

d = 8 eV). Electrons

are depleted in the blue region (background, ELF~0) and electrons are highly localized in the red

region (core region of each atom, ELF~1)

39

neighboring Ti atoms and oxygen vacancy. In the case of Ud = 8 eV, electronic charge is

transferred to the oxygen vacancy from the apical Ti atom. The electron density increases

between the equatorial Ti atoms and oxygen vacancy for Ud =8 eV as compared to U

d =7

eV, showing enhanced hybridization between equatorial Ti and vacancy site with

increasing Ud, and increased Coulomb repulsion between apical Ti and the vacancy site.

3.2.2 ATOMIC RELAXATION AROUND OXYGEN VACANCY

This significant change of the bonding character is strongly related to the atomic

relaxation around the vacancy. The atomic displacements are the measured distances of

the relaxed Ti atoms relative to the oxygen vacancy site. After relaxation, the three Ti

atoms surrounding the oxygen vacancy show outward relaxation from the vacancy, while

the oxygen atoms show inward relaxation. This relaxation trend in TiO2 is controlled by

the amount of charge transfer from the Ti ions, as has been also observed in other studies

[5]. However, there is a great discrepancy between the amount of displacement of

equatorial and apical Ti atoms for LDA+Ud with U

d = 7 and 8 eV. For U

d = 7 eV the

-8 -6 -4 -2 0 2 4 60

1

2

3

4

DO

S (

arb

. u

nit

s)

Energy (eV)

(a) Equatorial Ti

-8 -6 -4 -2 0 2 4 60

1

2

3

4

DO

S (

arb

. u

nit

s)

Energy (eV)

(b) Apical Ti

Fig. 3-6 Partial density of states of Ti atoms surrounding oxygen vacancy calculated by

LDA+Ud+U

p (U

d = 8 eV and U

p = 6 eV) (a) equatorial Ti and (b) apical Ti.

40

equatorial and apical Ti atoms move away from the vacancy by 0.3% and 0.6%,

respectively. Nevertheless, the Ud = 8 eV calculation shows that equatorial and apical Ti

atoms are relaxed by 5.9% and 6.6%, respectively, significantly larger than the atomic

displacement for Ud = 7 eV. This fact suggests that there are indeed serious limitations

for increasing Ud above 7 eV.

It is interesting to note that this additional defect state, the distorted conduction

bands and the enhanced relaxation observed for large Ud (8, 9 eV) disappear if U

p is

employed in addition to larger Ud values, in the LDA+U calculation. The partial density

of states of equatorial and apical Ti atoms, and ELF for LDA+Ud+U

p (U

d = 8 eV, U

p = 6

eV) are shown in Fig. 3-6, where density of states exhibits neither additional band gap

states nor distorted conduction bands as well as ELF shows similar bonding character

with LDA+Ud (U

d = 7 eV). These results lead us to the conclusion that the over

correction caused by the strong Ud can be compensated by U

p, and thus yielding a much

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

-40

-20

0

20

40

DO

S (

arb

. u

nit

s)

Energy (eV)

Fig. 3-7 Spin-polarized total density of states of rutile TiO2 2 x 2 x 3 supercell with isolated

single oxygen vacancy obtained by LDA+Ud+U

p (U

d = 8 eV and U

p = 6 eV). Fermi level is set

to 0 eV on the energy scale.

41

better agreement with experiment on the structural and electronic parameters of rutile

TiO2. In consideration of all the structural and electronic parameters, we used 8 eV and 6

eV for Ud and U

p for all calculations in the following part of this dissertation.

3.3 ELECTRON LOCALIZATION FUNCTION

3.3.1 ELECTRON DOPING BY AN ISOLATED SINGLE VACANCY

Fig. 3-7 shows the spin-polarized total density of states of TiO2 supercell with

isolated single oxygen vacancy. Here, and in the following part of this dissertation the

Fermi level position is set to 0 eV on density of states plots. The defect state is located

about 0.4 eV below the conduction band minimum. The defect state is fully occupied by

two electrons that were previously hybridized with O 2p orbitals and became localized on

Ti after the removal of O atom.

Fig. 3-8 shows the partial atomic density of states of each nearest neighboring Ti

and O atom surrounding oxygen vacancy. The defect state is observed only on the nearest

neighboring equatorial (,) and apical () Ti atoms indicating that two electrons are

highly localized on Ti 3d orbitals. No defect state formation is observed on the Ti atoms

() that are not in the immediate vicinity of the vacancy or on the O atoms (,). The

defect state, observed on the nearest neighboring Ti atoms, presents a minor difference in

the intensity of the defect state between equatorial and apical Ti atoms. This is due to the

asymmetry of the Ti-O bonds.

In addition, the defect state we obtained from this calculation is located around

0.4 eV below conduction band minimum. From the resistance switching point of view the

localization of two electrons and the position of defect state in the band gap can have

42

several implications. It is widely believed that when the system is in the low resistance

state, some kind of conductive path could be formed by the agglomeration of oxygen

vacancies as reported experimentally [6]. In such case, the resistance state depends on the

①

②

③

④

⑤

⑥

(a)

-8 -6 -4 -2 0 2 4 60.0

0.5

1.0

1.5

2.0

2.5

3.0

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

(b) Ti (①,②)

-8 -6 -4 -2 0 2 4 60.0

0.5

1.0

1.5

2.0

2.5

3.0

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

(c) Ti (③)

-8 -6 -4 -2 0 2 4 60.0

0.5

1.0

1.5

2.0

2.5

3.0

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

(d) Ti (④)

-8 -6 -4 -2 0 2 4 60.0

0.5

1.0

1.5

2.0

2.5

3.0

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

(e) O (⑤)

-8 -6 -4 -2 0 2 4 60.0

0.5

1.0

1.5

2.0

2.5

3.0

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

(f) O (⑥)

Fig. 3-8 (a) Schematics of rutile TiO2 (110) plane. The partial density of states of (b) and (c)

nearest neighboring Ti atoms, (d) Ti atom further from oxygen vacancy, (e) and (f) nearest

neighboring O atoms.

43

formation and disruption of this conductive path. Therefore, in the following we discuss

the implications of oxygen vacancy formation on the electron conduction.

It is well known that TiO2 exhibits n-type semiconducting behavior, because

oxygen vacancies effectively dope TiO2 with electrons. However, electrons are not likely

to be doped at room temperature since the defect state is too far from the conduction band

edge to dope electrons thermally to the conduction band. In comparison of doped-Si, the

donor or acceptor level is usually placed around 0.03 ~ 0.06 eV from the band edge,

hence dopants are easily excited even at room temperature. If we assume that those

electrons in the band gap state should be responsible for the low resistance of the system,

it may need a huge amount of thermal energy for electron transfer to the conduction band.

The electrons that are localized on the defect state are strongly confined in the vicinity of

the oxygen vacancy, and therefore their contribution is small to the overall electron

conduction that produces the low resistance state. Assuming that the formation of oxygen

vacancy path is responsible for the low resistance state, vacancies will be strongly

interacting with each other resulting in larger electronic charge redistribution. In such

case, there might be a possibility of electron hopping due to spreading out and splitting of

electron wave function, leading to the low resistance state of the system.

3.3.2 CHARGE DENSITY DISTRIBUTION

Fig. 3-9 shows the charge density difference contour plot of rutile TiO2 (110)

plane. The charge density difference is obtained by subtracting the charge density of

unrelaxed supercell from the charge density of the fully relaxed supercell. The oxygen

vacancy is labeled as “Vo” which is surrounded by three Ti atoms. The yellow dots

correspond to regions where charge density is increased, and the blue dots correspond to

44

regions where charge density is decreased. As shown in Fig. 3-9, 1st (,) and 2

nd ()

nearest neighboring atoms are Ti and next nearest neighboring atoms are O. There is an

asymmetry in original bond length between surrounding Ti atoms and O atom where

there is no vacancy. This asymmetry in bond length is the reason why there is almost no

charge density difference around 2nd

nearest neighboring Ti atom.

The charge redistribution due to the incorporation of oxygen vacancy is correlated

to the atomic relaxation, i.e. in the immediate vicinity of the vacancy, the Ti atoms move

away from the vacancy, whereas O atoms move towards the vacancy. In addition, the

charge density around three Ti atoms surrounding oxygen vacancy has been increased,

which was found in comparison with the charge density of the perfect supercell (without

vacancy) and defective supercell (with vacancy). This can be understood by the spatial

distribution of the two electrons which were hybridized to the O site that became the

Fig. 3-9 Charge density difference on rutile TiO2 (110) plane between the relaxed and

unrelaxed supercell when an oxygen vacancy is introduced. Δρ = ρ(relaxed) – ρ(unrelaxed).

45

vacancy. These electrons could be responsible for the increase of charge density of

surrounding Ti atoms leading to a lower oxidation state of these atoms.

Although the electronic charge redistribution is significant due to the breaking of

Ti-O bonds, the amount of atomic displacement of Ti atoms nearest to the vacancy is

very small in spite of the fact that the supercell was fully relaxed. In fact, 1st and 2

nd

nearest neighboring Ti atoms move away from the vacancy, by 0.011 Å and 0.002 Å

respectively. The difference of atomic movement between Ti atoms is due to the

asymmetry in the Ti-O bond length before the vacancy creation. The experimental Ti-O

bond length is 1.946 Å and 1.983 Å respectively. Therefore the relaxations are mostly

seen on the 1st nearest neighboring Ti atoms. The small displacement of nearest

neighboring atoms indicates that most of the energy relaxation is localized and conducted

by electronic charge redistribution rather than atomic displacements. This result suggests

the possibility that the electron hopping transport over oxygen vacancies may take place

Voequatorial Ti

apical Ti

Fig. 3-10 Band decomposed partial charge density distribution of TiO2 (110) plane with

isolated single oxygen vacancy.

46

if there are many oxygen vacancies within the limited area, such as vacancy clustering or

vacancy filament, resulting in strong interaction between vacancies.

In order to see the interaction between electrons, the most effective method is to

visualize the charge density profile. However it is not necessary to plot all charges

including those from the valence and conduction band. Since we are interested in the

behavior of electrons in only defect energy levels, total charge density must be

decomposed. In VASP calculation, charge density can be decomposed into each energy

level and the charge density within a certain range of energy level can be summed up. Fig.

3-10 represents the band decomposed partial charge density distribution of defect energy

states, which is drawn in the TiO2 (110) plane. As discussed above, electrons at defect

energy level is strongly localized around oxygen vacancy. But non-uniform distribution

of charges is observed indicating there is a preferential interaction between excess

electrons and adjacent Ti atoms.

3.4 OXYGEN VACANCY FORMATION ENERGY

Next, we calculated the oxygen vacancy formation energy. The formation energy

of neutral oxygen vacancy Evf (eV) is calculated as

Evf = E(VO) – E(TiO2) + µO, (3-4)

E(VO) is the total energy of the TiO2 supercell containing one neutral oxygen vacancy,

and E(TiO2) is the total energy of the perfect TiO2 crystal in the same supercell. µO is an

oxygen chemical potential. With pure LDA, the calculated total energy of oxygen

47

molecule is -10.66 eV. Assuming µO is half the total energy of an oxygen molecule, the

formation energy is 4.61 eV.

There have been a few theoretical attempts to calculate the formation energy of

oxygen vacancy, and the values obtained along with the approximations used are shown

in Table 3-1. Islam et al. [5] calculated the formation energy of oxygen vacancy to be

4.47 eV with the GGA which agrees well with the value of 4.44 obtained by Cho et al.

[7] obtained with LDA. Bouzoubaa et al. [8] also reported that the formation energy of

oxygen vacancy is 4.52 eV with GGA. In contrast, using LDA, Hameeuw et al. [9]

obtained the formation energy of oxygen vacancy of 6.03 eV. Shu et al. [10] and Iddir et

al. [11] found that the GGA bulk vacancy formation energies are 5.1 eV and 4.93 eV,

respectively. Therefore, there is considerable controversy about the calculation of the

Table 3-1. Summary of theoretical results of oxygen vacancy formation energy in rutile TiO2.

Method Code Potential Evf (eV) Reference

GGA CRYSTAL03 4.47 Islam et al. [5]

PW1PW CRYSTAL03 5.08 Islam et al. [5]

LDA VASP PAW 4.44 Cho et al. [7]

GGA VASP US-PP 4.52 Bouzoubaa et al. [8]

LDA Quantum-ESPRESSO US-PP 6.03 Hameeuw et al. [9]

GGA VASP PAW 5.10 Shu et al. [10]

GGA VASP US-PP 4.93 Iddir et al. [11]

HSE VASP PAW 5.50 Janotti et al. [14]

LDA VASP PAW 4.61 This work

48

formation energy of the oxygen vacancy in TiO2. Experimental results of Kofstad et al.

[12] and Marucco et al. [13] showed that the formation enthalpy of doubly ionized

oxygen vacancy is 4.55 and 4.57 eV, respectively. Additionally Marucco et al. observed

that full ionization energies of neutral oxygen vacancy are not greater than 0.30 eV.

Therefore the experimental formation energy of neutral oxygen vacancy should be less

than 4.55 eV by the amount of 0.3 eV.

The formation energies of oxygen vacancy obtained from the LDA+U

calculations for various Ud and U

p values, are presented in Table 3-2. U

p corrections are

included in the calculation of the energy of oxygen molecule. The formation energies

with LDA+U calculations are slightly larger than that with the conventional LDA.

Although there are small discrepancies in vacancy formation energies, these values are

almost independent of U parameters, since equation (3-4) includes the reference energy

for the oxygen molecule, which is corrected by Up parameters, and the U

d and U

p

corrections are applied the total energy of TiO2 supercell. It is worth noting that absolute

values of the formation energies calculated with the LDA+U method, as a function of U,

are arbitrarily determined. Therefore, it is more practical to examine the transition states

between different charged states of the vacancies than the absolute values of the

Table 3-2. Summary of formation energy of oxygen vacancy with various values of Ud and U

p.

Ud (eV) 3 4 5 6 7 8 9

Evf (eV) 5.41 5.68 6.05 6.10 6.11 5.83 5.66

Up(U

d=8 eV) 3 4 5 6 7 8 9

Evf (eV) 6.21 6.26 6.31 6.37 6.43 6.50 6.58

49

formation energies.

3.5 POSITIVELY CHARGED OXYGEN VACANCY

3.5.1 DENSITY OF STATES

In real TiO2, it is well known that positively charged oxygen vacancies effectively

dope TiO2 with electrons, resulting in n-type semiconducting behavior [15]. As shown in

Fig. 3-11, we calculated the density of states of rutile TiO2 with positively charged

oxygen vacancies. LDA+Ud+U

p, where U

d = 8 eV and U

p = 6 eV, is used in this

calculation. In the case of singly charged oxygen vacancy (Vo1+

), the spin-down defect

state shift towards the conduction band minimum, whereas both spin-up and down defect

states are observed below the conduction minimum for doubly charged oxygen vacancy

(Vo2+

). This asymmetric density of states particularly in Vo1+

may result in the

ferromagnetic effect of undoped rutile TiO2. Several experiments and theoretical

investigations reported that the oxygen vacancy induces a magnetic moment in rutile

-8 -6 -4 -2 0 2 4 6-60

-40

-20

0

20

40

60

(a) Vo1+

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

-6 -4 -2 0 2 4 6 8-60

-40

-20

0

20

40

60

(b) Vo2+

DO

S (

arb

. u

nit

s)

Energy (eV)

EF

Fig. 3-11 Total density of states of rutile TiO2 with (a) Vo1+

, and (b) Vo2+

calculated by

LDA+Ud+U

p (U

d = 8 eV, U

p = 6 eV).

50

TiO2 [16, 17].

3.5.2 ELECTRON LOCALIZATION AND ATOMIC RELAXATION EFFECTS

The different behavior of the electronic states in different charge state of the

oxygen vacancy is linked to the atomic displacement of nearest neighboring atoms

surrounding oxygen vacancy. Fig. 3-12 shows ELF and structural relaxations of TiO2

with charged oxygen vacancy. The neighboring Ti atoms show outward relaxation 5.8 ~

7.6% for Vo1+

and 12% for Vo2+

, respectively, which are much larger relaxations than

0.4% outward relaxation in neutral oxygen vacancy. Similar results were observed by

Janotti et al. for ZnO [18]. With the removal of electrons from oxygen vacancies a

weaker Coulomb interaction between Ti atoms and the vacancy site is observed, resulting

in larger displacement of the neighboring Ti atoms. As a result, the empty defect states

shift towards the bottom of the conduction band.

Fig. 3-12 Electron localization function and structural relaxations around charged oxygen

vacancy in LDA+Ud+U

p (U

d = 8 eV, U

p = 6 eV). (a) Vo

1+, and (b) Vo

2+.

51

3.5.3 STABILITY OF CHARGED OXYGEN VACANCY

To determine the relative stability of the various charge states of oxygen vacancy,

we calculated the vacancy formation energy of the positively and negatively charged

oxygen vacancy using the following formula:

Evf(Vq) = E(V

q) – E(TiO2) +1/2E(O2) + µO + qEF, [14] , (3-5)

where E(Vq) is the total energy of the supercell containing a vacancy in charge state q and

E(TiO2) is the total energy of perfect supercell. µO is the chemical potential of oxygen

atom, which is a variable. This chemical potential must satisfy the growth condition of

TiO2, namely µTi + 2µO = ∆HfTiO2 = -11.1 eV. In the extreme O-rich limit, µO = 0. The Ti-

rich limit is constrained by the formation of Ti2O3; 2µTi + 3µO ≤ ∆HfTi2O3 = 19.1 eV, which

gives µO = 3.17 eV. The Fermi level EF is referenced to the valence band maximum (EF =

0 eV) and EF term includes the energy of the valence band maximum in the

stoichiometric system and core-level electrostatic potential alignment between the defect

supercell and the perfect crystal.

The calculated vacancy formation energies in unrelaxed and relaxed TiO2

supercell as a function of Fermi level are shown in Fig. 3-13(a) and (b). Structural

relaxation lowers the formation energy of the neutral vacancy (Vo) and singly charged

vacancy (Vo1+

) by amount of 0.23eV and 1.28eV, respectively. For the doubly charged

vacancy (Vo2+

), relaxation lowers the formation energy by 2.95 eV, which is consistent

with the different amount of lattice relaxation with respect to the charge state as described

above. We find the transition levels ε(2+/+) and ε(1+/0) to be located in the band gap

52

around 0.7 eV and 0.5 eV, respectively below the conduction band minimum. The lowest

formation energy of the doubly charged vacancy (Vo2+

) for a wide range of Fermi level

accounts for the observed n-type semiconducting behavior in TiO2.

Janotti et al. [14] have studied the vacancy formation energy in rutile TiO2 using

hybrid functional approximations (HSE). Similar behavior of lattice relaxation was found

for the various charge states of oxygen vacancy. According to Janotti‟s work, Vo2+

has

the lowest formation energy for all values of the Fermi level and the transition levels

ε(2+/+) and ε(1+/0) are located above the conduction band minimum. This difference is

attributed to the electronic correlation approximations, and remains to be elucidated

further for the LDA+U and HSE approximations. Although there is a small difference,

our results for the formation of the charged vacancies essentially agree with the results in

several previous studies [11, 19, 20].

The vacancy formation energy of rutile TiO2 in the limit of Ti-rich and O-rich are

shown in Fig. 3-13(b) and (c). The formation energy of the various charge states of the

vacancy in Ti-rich condition is lower than that in O-rich condition. The oxygen vacancy

0 1 2 3-2

0

2

4

6

8

Form

ati

on

En

ergy

(eV

)

(a) unrelaxed, Ti-rich

Vo

Vo1+

Vo2+

Fermi Level (eV)

0 1 2 3-2

0

2

4

6

8

(b) relaxed, Ti-rich

Vo

Vo1+

Vo2+

Fermi Level (eV)

0 1 2 3-2

0

2

4

6

8

(c) relaxed, O-rich

Vo

Vo1+

Vo2+

Fermi Level (eV)

Fig. 3-13 Oxygen vacancy formation energy in rutile TiO2 as a function of Fermi level. (a)

unrelaxed in Ti-rich condition, (b) relaxed in Ti-rich condition, and (c) relaxed in O-rich

condition.

53

concentration increases when the chemical potential of oxygen is lowered, due to its

lower formation energy in Ti-rich (O-poor) condition. Several experimental studies [21,

22]observed that electrical conductivity of TiO2 was found to decrease with increasing

oxygen partial pressure, i.e. the oxygen vacancy concentration might be crucial for fine-

tuning the electrical properties of rutile TiO2.

54

REFERENCES

[1] J. Chen, L.-B. Lin, and F.-Q. Jing, “Theoretical study of F-type color center in rutile

TiO2,” Phys. Chem. Solids, vol. 62, pp. 1257-1262, Apr 2001

[2] E. Cho, S. Han, H.-S. Ahn, K.-R. Lee, S. Kim, and C. Hwang, “First-principles study

of point defects in rutile TiO2-x,” Phys. Rev. B, vol. 73, pp. 193202, May 2006

[3] M. Ramamoorthy, R. D. King-Smith, and D. Vanderbilt, “Defects on TiO2 (110)

surfaces,” Phys. Rev. B, vol. 49, pp. 7709-7715, Mar 1994

[4] A. D. Becke, and K. E. Edgecombe, “A simple measure of electron localization in

atomic and molecular systems,” J. Chem. Phys., vol. 92, pp. 5397, Jan 1990

[5] M . M. Islam, T. Bredow, and A. Gerson, “Electronic properties of oxygen-deficient

and aluminum-doped rutile TiO2 from first principles,” Phys. Rev. B, vol. 76, pp.

045217, Jul 2007

[6] M. Janousch, G. I. Meijer. U. Staub, B. Delly, S. F. Karg, and B. P. Andreasson,

“Role of oxygen vacancies in Cr-doped SrTiO3 for resistance-change memory,” Adv.

Mater., vol. 19, pp. 2232-2235, 2007

[7] E. Cho, S. Han, H.-S. Ahn, K.-R. Lee, S. Kim, and C. Hwang, “First-principles study

of point defects in rutile TiO2-x,” Phys. Rev. B, vol. 73, pp. 193202, May 2006

[8] A. Bouzoubaa, A. Markovits, M. Calatayud, and C. Minot, “Comparison of the

reduction of metal oxide surfaces: TiO2-anatase, TiO2-rutile and SnO2-rutile,” Surf.

Sci., vol. 583, pp. 107-117, May 2005

55

[9] K. Hameeuw, G. Cantele, D. Ninno, F. Trani, and G. Iadonisi, “Influence of surface

and subsurface defects on the behavior of the rutile TiO2 (110) surface,” Phys. Stat.

Sol., vol. 203, pp. 2219 Jul 2006

[10] Da-Jun Shu, Shu-Ting Ge, Mu Wang, and Nai-Ben Ming, “Interplay between

external strain and oxygen vacancies on a rutile TiO2 (110) surface,” Phys. Rev.

Lett., vol. 101, pp. 116102 Sep 2008

[11] H. Iddir, S. Öğüt, P. Zapol, and N. D. Browning, “Diffusion mechanisms of native

point defects in rutile TiO2: Ab initio total-energy calculations,” Phys. Rev. B, vol.

75, pp. 073203 Feb 2007

[12] P. Kofstad, Nonstoichometry, Diffusion, and Electrical Conductivity in Binary Metal

Oxides (Wiley, New York, 1972), Chap. 8

[13] J. F. Marucco, J. Gautron, and P. Lemasson, “Thermogravimetric and electrical

study of non-stoichiometric titanium dioxide TiO2-x between 800 and 1100oC,” J.

Phys. Chem. Solids, vol. 42, pp. 363, Aug 1981

[14] A. Janotti, J. B. Varley, P. Rinke, N. Umezawa, G. Kresse, and C. G. Van de Walle,

“Hybrid functional studies of the oxygen vacancy in TiO2,” Phys. Rev. B, vol. 81,

pp. 085212, Feb 2010

[15] E. G. Eror, “Self-compensation in niobium-doped TiO2,” J. Solid State Chem., vol.

38, pp. 281-287, Jan 1981

56

[16] D. Kim, J. Hong, Y. Park, and K. Kim, “The origin of oxygen vacancy induced

ferromagnetism in undoped TiO2,” J. Phys.: Condens. Matter, vol. 21, pp. 195405

May 2009

[17] N. H. Hong, J. Sakai, N. Poirot, and V. Brizé, “Room-temperature ferromagnetism

observed in undoped semiconducting and insulating oxide thin films,” Phys. Rev. B,

vol. 73, pp. 132404, Apr 2006

[18] A. Janotti and C. G. Van de Walle, “Oxygen vacancies in ZnO,” Appl. Phys. Lett.,

vol. 87, pp. 122102, Sep 2005

[19] J. M. Sullivan and S. C. Erwin, “Theory of dopants and defects in Co-doped TiO2

anatase,” Phys. Rev. B, vol. 67, pp. 144415, Apr 2003

[20] S. Na-Phattalung, M. F. Smith, K. Kim, M.-H Du, S.-H Wei, S. B. Zhang, and S.

Limpijumnong, “First-principles study of native defects in anatase TiO2,” Phys. Rev.

B, vol. 73, pp. 125205, Mar 2006

[21] R. K. Sharma, M. C. Bhatnagar, and G. L. Sharma, “Effect of Nb metal ion in TiO2

oxygen gas sensor,” Appl. Sur. Sci., vol. 92, pp. 647-650, Oct 1996

[22] J. Yahia, “Dependence of the electrical conductivity and thermoelectric power of

pure and Aluminum-doped rutile on equilibrium oxygen pressure and temperature,”

Phys. Rev., vol. 130, pp. 1711-1719, Jun 1963

57

CHAPTER 4: DI-VACANCY

Oxygen vacancies are also found to exist in the form of clustering such as di-vacancy, tri-

vacancy, or vacancy clusters. In this chapter, we discuss the effect of di-vacancy on the

electronic structure of TiO2. Possible relative positions between the vacancies in a di-

vacancy are investigated to find the most stable di-vacancy configuration. The formation

of Ti-Ti bonding that can affect the conductivity of TiO2 is discussed. The density of

states, electron localization function, and band decomposed partial charge density are

presented.

4.1 INTERATOMIC DISTANCE EFFECTS FOR DI-VACANCY

58

4.1.1 STABLE DI-VACANCY CONFIGURATION

We first constructed several di-vacancy configurations with various relative

positions of vacancies, in order to investigate the interaction between two neutral oxygen

vacancies. Fig 4-1 shows eight different di-vacancy structures we have explored

depending on the distance between two vacancies. 3 x 3 x 4 supercell of TiO2 that

contains 72 Ti and 144 O atoms is used in this calculation. Di-vacancy structures can be

divided into two categories, two vacancies in the same (110) plane, and two vacancies in

the different (110) plane. We call the former „in-plane di-vacancy‟ and the latter „out-of-

plane di-vacancy‟ in this dissertation.

(a) 2.53Å (b) 3.97 Å (c) 6.50 Å (d) 10.46 Å

(e) 3.33 Å (f) 4.59 Å (g) 5.92 Å (h) 6.50 Å

Fig. 4-1 Schematic configurations of eight different di-vacancy structures. (a), (b), (c), (d). In-

plane di-vacancy, and (e), (f), (g), (h). Out-of-plane di-vacancy. The inter-vacancy distance is

shown.

59

To determine the most stable configuration of two vacancies, the interaction

energy (Eint) is calculated with respect to the reference structure containing two isolated

oxygen vacancies, where two vacancies are placed at large distance from each other,

using the following formula:

Eint = E(2Vo) + E(TiO2) – 2E(1Vo), [1] , (4-1)

where E(TiO2) is the total energy of the perfect supercell and E(Vo) is the total energy of

the supercell including a number of vacancies as indicated in the parenthesis. Fig. 4-2

shows calculated interaction energies of two oxygen vacancies as a function of the inter-

2 4 6 8 10 12-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

Inte

ra

cti

on

En

erg

y (

eV

)

Inter vac. distance (Å )

(a)

(b)

(c)

(d)

(e) (f) (g)(h)

Fig. 4-2 Interaction energy between two oxygen vacancies as a function of inter-

vacancy distance. (a), (b), (c), and (d) represent in-plane di-vacancy and (e), (f), (g),

and (h) represent out-of-plane di-vacancy.

60

vacancy distance. While out-of-plane di-vacancy structures, denoted as (e), (f), (g), and

(h), show nearly zero interaction energies, in-plane di-vacancy structures, denoted as (a),

(b), (c) and (d), show negative values for the interaction energies. The in-plane di-

vacancy structure is more stable than isolated vacancy configurations, and the interaction

energy is almost vanishing when the two vacancies are not located in the same (110)

plane. Therefore in-plane di-vacancy structures are likely to be formed if oxygen

vacancies are close enough, which might be expected to behave in a different way from

the isolated single vacancy configuration. This is associated with the fact that (110) plane

in rutile TiO2 is the bonding plane between Ti and O. Therefore the relative position of

the oxygen vacancies and the physical distance between the vacancies, both are important

in the determination of the interaction between vacancies.

As shown in Fig. 4-2, the lowest interaction energy is observed for the structure

(a), called equatorial di-vacancy, in which two vacancies are nearest neighboring with

2.53Å of inter-vacancy distance, that is shorter than that of apical di-vacancy, 3.97 Å .

The apical di-vacancy structure is denoted as structure (b). This is in a good agreement

with a previous work [2] that has shown that the total energy of the apical di-vacancy is

higher compared to that of the equatorial di-vacancy.

4.1.2 INTERACTION BETWEEN TWO VACANCIES IN THE (110) PLANE

The strong interaction between oxygen vacancies can also be inferred by

investigating the density of states. Fig. 4-3 shows the total density of states of four

different in-plane di-vacancy structures, in which in-gap defect states are observed in all

cases. In contrast to isolated single oxygen vacancy, defect states are considerably

dispersed, and the position of defect energy levels is significantly altered depending on

61

the configuration of two vacancies. This is due to the strong interaction between the

electrons localized in the two oxygen vacancies sites.

Defect energy levels become more localized as the distance between two

vacancies increases, indicating the strongest interaction between defect energy levels in

(a)-8 -6 -4 -2 0 2 4 6

-150

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

Energy (eV)

(b)-8 -6 -4 -2 0 2 4 6

-150

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

Energy (eV)

(c)-8 -6 -4 -2 0 2 4 6

-150

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

Energy (eV)

(d)-8 -6 -4 -2 0 2 4 6

-150

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

Energy (eV)

Fig. 4-3 Total density of states of in-plane di-vacancy structures.

62

equatorial and apical di-vacancy. It should be noted that defect states in both equatorial

and apical di-vacancy configurations are much shallower than those corresponding to an

isolated vacancy, which induces a defect state at about 0.4 eV below conduction band

minimum [3]. All these defect states are occupied by electrons. Thus it is believed that it

is likely that vacancy clusters rather than isolated vacancies play an important role in the

n-type conductivity of rutile TiO2.

On the other hand, an overlap of defect energy levels is observed for the out-of-

plane di-vacancy, resembling the density of states of an isolated single oxygen vacancy.

This indicates that there is a negligible interaction between oxygen vacancies in out-of-

plane di-vacancy configuration, and is consistent with the result of the interaction energy

calculation as shown in Fig. 4-2. The density of states of two different out-of-plane di-

(a)-8 -6 -4 -2 0 2 4 6

-150

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

Energy (eV)

(b)-8 -6 -4 -2 0 2 4 6

-150

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

Energy (eV)

Fig. 4-4 Total density of states of out-of-plane di-vacancy structures.

63

vacancy structures is shown in Fig. 4-4. The position of the defect states is very similar to

that of isolated single oxygen vacancy. Therefore we conclude that if two vacancies are

not located in the same (110) plane, the interaction between vacancies can be negligible.

In addition, while the electron spins distribution corresponding to the defect

energy level is symmetrical in out-of-plane di-vacancy structures, in-plane di-vacancy

structures show asymmetric spin distribution. This implies that there is a strong repulsion

between excess electrons therefore electrons tend to occupy different energy levels with

the same spin direction. This may induce ferromagnetic characteristics in TiO2 in which

oxygen vacancies play an important role [4-6].

4.2 TI-TI BONDS IN EQUATORIAL DI-VACANCY

In case of the equatorial di-vacancy, we found that its particular geometry of the

vacancy configuration results in change in the bonding character of Ti atoms around

those oxygen vacancies. Fig. 4-5 displays the electron localization function (ELF) of an

(a) (b)

Fig. 4-5 Electron localization function of (a) isolated single oxygen vacancy, and (b)

equatorial di-vacancy.

64

isolated single vacancy and the equatorial di-vacancy structure showing the strength of

electron localization. The (110) plane is chosen since all Ti-O bonds are in this plane. As

described in chapter 3, electron localization function has a value between 0 and 1 that

gives account of the type of bonding preferred. The perfectly localized region is given by

the value of 1, showing mostly covalent bonding character. The value of 0.5 corresponds

to an electron gas with metallic nature, and the values between 0 and 0.5 indicate regions

of low electron density dominated by strong ionic bonding character.

The most noticeable difference between an isolated single vacancy and di-

vacancy is the modification of the bonding character around vacancies. While an isolated

single vacancy structure shows mostly ionic bonding character with a weak metallic

nature between Ti and electrons localized in the vacancy site, mostly strong metallic

bonds are formed in an equatorial di-vacancy. These metallic bonds connect all Ti atoms

Ti

O

eg

Ti

O

t2g

(a) (b)

Fig. 4-6 Schematic pictures of TiO2 octahedral structure showing the direction of

Ti 3d orbitals (a) eg orbitals, and (b) t2g orbitals.

65

surrounding two oxygen vacancies, thus metallic conduction could be possible among

those surrounding Ti atoms.

In addition, it should be noted that the shape of the electron localization function

of the two equatorial Ti atoms lying in between two vacancies is considerably different

from that in an isolated single vacancy structure. This implies that there is a remarkable

change in the electron density distribution around Ti atoms and it affects the valence state

of Ti atoms. Compared to an isolated single vacancy, two equatorial Ti atoms show a

directional change in electron density distribution from diagonal direction toward oxygen

atoms to either vertical or horizontal direction, while oxygen atoms still maintain their

bonding electron density in a spherical manner. This indicates that excess electrons in

equatorial di-vacancy structure are withdrawn from Ti eg orbitals and reside in t2g orbitals

resulting in the depletion of electrons in eg orbitals. The schematic pictures of Ti 3d

(a) (b)

Fig. 4-7 Charge density map of TiO2 and LiTiO2. (a) Difference electron density map

constructed from all valence densities of TiO2, and (b) Difference electron density

map constructed of Li electron density in LiTiO2. [7]

66

orbital in rutile TiO2 octahedral structure are shown in Fig. 4-6. Note that eg orbitals are

pointing toward oxygen atoms. The depletion of electron density in eg orbitals results in

the loss of covalent interaction between Ti (eg) and O (2p) orbitals, and then increase the

metallic interaction between two equatorial Ti (t2g) building up Ti-Ti bonds. The

direction of this bond is parallel to [001] direction (c-axis).

This kind of orbital change and the formation of Ti-Ti bonds have been reported

earlier in LiTiO2, which is known as one of the superconducting materials [7-9]. Fig. 4-7

shows charge density map of pure TiO2 and Li-intercalated-TiO2. They displayed the

difference electron valence density averaged over the whole region for pure TiO2 as

shown in Fig. 4-7(a), and for Li-intercalated-TiO2 they displayed the difference electron

band density constructed only from the Li electron density as shown in Fig. 4-7(b). An

important feature in this figure is the fact that after the intercalation of Li the covalent Ti

(eg) - O (2p) bonding is considerably diminished. It shows that electrons in Li sites are

depleted and transferred to Ti atoms building up the metal-to-metal bonds of t2g-t2g type.

This Ti-Ti network, which is composed of degenerate t2g-like orbitals, represents the

electronic foundation of superconductivity observed in LiTiO2. Therefore the transfer of

electron density from Li atoms to Ti atoms enables the formation of Ti-Ti network and

the superconductivity of LiTiO2. The Li effect on this structure is very similar to that of

oxygen vacancies in rutile TiO2. Therefore the fact that Ti-Ti bonds could be formed by

two adjacent oxygen vacancies gives us some clues to explain the metallic “ON”-state

conduction mechanism in TiO2.

67

4.3 CHARGE OCCUPANCY OF THE t2g ORBITAL OF TI

4.3.1 PARTIAL CHARGE DENSITY AND ORBITAL OCCUPANCY

The formation of Ti-Ti network is also observed in the band decomposed charge

density profile as shown in Fig. 4-8 that is plotted by iso-surface charge density with an

iso-value of 0.1 e/Å3. The band decomposed charge density represents the partial charge

density distribution corresponding to the defect states within the band gap. As shown in

Fig. 4-5(b), charge density of excess electrons in the defect states connects all Ti atoms

surrounding oxygen vacancies resulting in metallic bonds among Ti atoms. This indicates

that defect states are not localized on certain Ti atom but on all four Ti atoms surrounding

two vacancies, implying that electrons can be easily transported among these Ti atoms.

Equatorial Ti

Apical Ti

Vo

0.1e/Å 3

Fig. 4-8 Iso-surface of band decomposed partial charge density of di-vacancy in

TiO2. Iso-surface is drawn with an iso-value of 0.1e/Å3.

68

The change of orbital occupation from eg to t2g orbital in Ti atoms shown in the

electron localization function is also observed in the partial charge density distribution.

Based on this result, it can be assumed that if there are many di-vacancy or vacancy

clusters in bulk TiO2, these metal-to-metal bonds are likely to substantially improve the

conductivity of TiO2.

The analysis of the projected charge corresponding to the wave function of the

each defect state was carried out to confirm that occupied orbital change in Ti atoms

observed in the electron localization function and partial charge density distribution, as

shown in Fig. 4-9. The partial charge density distribution of each defect energy state is

also displayed. Due to four excess electrons from the two oxygen vacancies, there are

four defect states in which numbers (1 ~ 4) represent each defect energy level. The cross-

shaped charge density distribution is found in defect state (2) in which most charges in

dxy

pz

px

py

Pa

rti

al

ch

arg

e (

arb

. u

nit

s)

s

dx

2y

2dxz

dz

2dyz

(1)

(3)

(4)

(2)

Fig. 4-9 Schematic diagram of orbital occupancy of charges in four defect states

in di-vacancy structure.

69

the 3d orbital are occupied by dyz and dxz orbital that are belong to t2g. Hence, partial

occupancy in t2g orbital accounts for the cross-shaped charge distribution to the equatorial

Ti atoms.

4.3.2 PREFERENTIAL DISTRIBUTION OF EXCESS ELECTRONS

The formation of equatorial di-vacancy modifies the charge distribution around

vacancies including adjacent Ti atoms. We found that this charge redistribution is not

uniform implying that there is a preferential interaction between Ti atoms and the excess

electrons. Fig. 4-10 shows the partial charge density distribution of equatorial di-vacancy

structure. The charge density in equatorial Ti atoms is higher than that in apical Ti atoms.

This means that excess electrons are more localized on equatorial Ti atoms than apical Ti

atoms. The partial density of states of each Ti atom is in a good agreement with this result.

Fig. 4-11 demonstrates that more defect states are found in equatorial Ti atoms with

Equatorial Ti

Apical Ti

Vo

Fig. 4-10 Band decomposed partial charge density distribution of di-vacancy

in TiO2 (110).

70

higher intensity than apical Ti atoms.

In order to verify the preferential distribution of excess electrons, we calculated

the atomic charge density. Since electronic charge density is continuous, it is not clear

how one should partition electrons amongst fragments of the system such as atoms or

molecules. We integrated the atomic charge density using the so-called Bader charge

analysis [10]. Bader analysis is an intuitive way of dividing molecules into atoms based

on the electronic charge density. It uses what are called zero flux surfaces, a 2-D surface

on which the charge density is at minimum perpendicular to the surface, to divide the

atomic contibutions. Typically in molecular systems, the charge density reaches a

minimum between atoms and this is natural place to separate atoms from each other. As a

result, the charge density of two equatorial Ti atoms is increased by 0.73 e/atom while

two apical Ti atoms show the increase of charge density by 0.33 e/atom.

It is known that the bonding character between Ti and O in TiO2 is mostly ionic

with partial covalent nature. However Ti atom is likely to lose its ionic properties if

electrons are supplied to Ti atom. Therefore additional gain in electronic charges makes

-8 -6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Energy (eV)-8 -6 -4 -2 0 2 4 6

-5

-4

-3

-2

-1

0

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Energy (eV) (a) Equatorial Ti (b) Apical Ti

Fig. 4-11 Partial density of states of (a) equatorial Ti, and (b) apical Ti.

71

equatorial Ti atoms more metallic than normal Ti4+

. As a result, metallic conduction

between those two equatorial Ti atoms could be achieved. This fact suggests that the

vacancy clusters are likely to increase the conductivity of TiO2, and the equatorial di-

vacancy configuration that has the shortest inter-vacancy distance may increase the

conductivity.

72

REFERENCES

[1] S. Park, H.-S. Ahn, C.-K. Lee, H. Kim, H. Jin, H.-S. Lee, S. Seo, J. Yu, and S. Han,

“Interaction and ordering of vacancy defects in NiO,” Phys. Rev. B, vol. 77, pp.

134103, Apr 2008

[2] E. Cho, S. Han, H.-S. Ahn, K.-R. Lee, S. Kim, and C. Hwang, “First-principles study

of point defects in rutile TiO2-x,” Phys. Rev. B, vol. 73, pp. 193202, May 2006

[3] S. G. Park, B. Magyari-Köpe, and Y. Nishi, “Electronic correlation effects in reduced

TiO2 within the LDA+U method,” Phys. Rev. B, vol. 82, pp. 115109, Sep 2010

[4] N. Hoa Hong, J. Sakai, N. Poirot, and V. Brize, “Room temperature ferromagnetism

observed in undoped semiconducting and insulating oxide thin films,” Phys. Rev. B,

vol. 73, pp. 132404, Apr 2006

[5] S. D. Yoon, Y. Chen, A. Yang, T. L. Goodrich, X. Zuo, D. A. Arena, K. Ziemer, C.

Vittoria, and V. G. Harris, “Oxygen-defect-induced magnetism to 880K in

semiconducting anatase TiO2-δ films,” J. Phys.: Condens. Matter, vol. 18, pp. L355,

Jun 2006

[6] A. K. Rumaiz, B. Ali, A. Ceylan, M. Boggs, T. Beebe, and S. I. Shah, “Experimental

studies on vacancy induced ferromagnetism in undoped TiO2,” Solid State

Communication, vol. 144, pp. 334-338, Sep 2007

[7] L. Benco, J.-L. Barras, and C. A. Daul, “Theoretical study of the intercalation of Li

into TiO2 structures,” Inorg. Chem., vol. 38, pp. 20-28, 1999

73

[8] J. A. Campa, M. Velez, C. Cascales, E. P. Gutierrez Puebla, M. A. Monge, “Cystal

growth of superconducting LiTi2O4,” J. Crystal Growth, vol. 142, pp. 87, 1994

[9] E. Moshopoulou, P. Bordet, A. Sulpice, and J. Capponi, “Evolution of structure and

superconductivity of Li1-xTi2O4 single crystals without Ti cation disorder,” J. Physica

C, vol. 235, pp. 747-748, 1994

[10] W. Tang, E. Sanville, and G. Henkelman, “A grid-based Bader analysis algorithm

without lattice bias,” J. Phys.: Condens. Matter, vol. 21, pp. 084204, Feb 2009

74

CHAPTER 5: EFFECT OF VACANCY ORDERING ON THE

“ON”-STATE CONDUCTION

In this chapter, we investigated the effect of vacancy ordering on the “ON”-state

conduction in TiO2 during resistance switching. Two different direction of vacancy

ordering, [110] and [001], are examined. As in the di-vacancy study, electron localization

function, density of states, and charge density distribution are shown. In order to judge

the stability of vacancy-ordered structure, the vacancy formation energies are calculated

and compared to other vacancy structures. We performed the calculation of energy band

structure and discussed the effect of vacancy ordering on the resistance of TiO2.

75

5.1 MULTI VACANCY CONFIGURATIONS

We increased the number of oxygen vacancies along two different directions in

TiO2 (110) plane, [110] and [001], as shown in Fig. 5-1. The electron localization

function is first investigated as shown in Fig. 5-2. As the number of oxygen vacancies is

increased, it is noted that the strength of the electron localization on vacancy sites is

getting weaker, and then in case of vacancy-ordered structure along [001] direction it

seems that electrons are no longer localized on vacancy site. This indicates that more

charges are localized in the adjacent Ti atoms and contribute to increase the metallic

properties of those Ti atoms. It is noticeable that the cross bar shaped electron

localization function is also observed in [001] vacancy-ordered structure in equatorial Ti

2Vo 3Vo

5Vo 6Vo

4Vo

8Vo

[110] [001]

Vo

(a) (b)

Fig. 5-1 Schematic configurations of increasing number of oxygen vacancies along two

directions in TiO2 (110). (a) Vacancy ordering in [110], and (b) vacancy ordering in

[001].

76

atoms. As discussed in di-vacancy study, this is due to the fact that some valence

electrons in equatorial Ti atoms are occupied by t2g orbitals, resulting in Ti-to-Ti bonds

involving overlap of t2g-t2g type of orbitals. Based on electron localization function,

therefore, it is likely that vacancy-ordered structure, specially the [001] vacancy chain

will have a significant contribution to electron transport.

5.2 FORMATION OF A CONDUCTIVE CHANNEL BY OXYGEN VACANCY

ORDERING

5.2.1 DENSITY OF STATES OF THE VACANCY-ORDERED STRUCTURE

Fig. 5-3 shows the total density of states of two vacancy-ordered structures along

[110] and [001] respectively. Compared to an isolated single vacancy or di-vacancy,

[110] [001]

3Vo

5Vo 6Vo

4Vo

8Vo

Vo

2Vo

(a) (b)

Fig. 5-2 Electron localization function of structures shown in Fig. 5-1. The number of

oxygen vacancies is increased along two directions. (a) Vacancy ordering in [110], and

(b) vacancy ordering in [001].

77

more defect states are found in the band gap. It seems that defect states for the [001]

vacancy-ordered structure are covering the entire area of the band gap, therefore it might

be possible for electrons in the valence band to be transported to the conduction band

lowering the resistance of the system considerably.

Fig. 5-4 shows the partial density of states of nearest neighboring equatorial and

apical Ti atoms in both [110] and [001] vacancy-ordered structures. [110] vacancy-

ordered structure shows comparable amount of defect states in both equatorial and apical

Ti atoms. In contrast, in [001] vacancy-ordered structure, the defect states are distributed

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

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

E - EF (eV)

(a)

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

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

E - EF (eV)

(b)

Fig. 5-3 Total density of states of two vacancy ordered structures. (a) Vacancy-

ordered structure along [110], and (b) vacancy-ordered structure along [001].

78

throughout the band gap with a partial overlap between them, and we found that most of

the defect states originate from equatorial Ti atoms, implying that most of the electrons

are transferred to equatorial Ti atoms. This indicates that the geometric configuration of

oxygen vacancies significantly affects the redistribution of electrons. As a consequence,

this preferential interaction between electrons and equatorial Ti atoms observed in [001]

vacancy-ordered structure will make equatorial Ti atoms more metallic by increasing the

charge density, thus metallic conduction through these Ti atoms can be occurred by

defect assisted tunneling, which may be the explanation of the “ON”-state conduction in

TiO2.

5.2.2 CHARGE DENSITY DISTRIBUTION AND BADER ANALYSIS

The integration of the electronic charge density of Ti atoms can be obtained from

the Bader charge analysis. From this analysis, we found that there is a significant

difference in charge density increase between equatorial and apical Ti atoms in [001]

vacancy-ordered structure. In this study, we used pseudopotentials, which means that

only limited valence electrons are considered because core electrons hardly interact with

forming and breaking bonds. 12 valence electrons in Ti (3s23p

63d

24s

2) and 6 electrons in

O (2s22p

4) are considered in the calculation. Therefore, if bonding in TiO2 is completely

ionic, each Ti atom loses 4 electrons and each O atom gains 2 electrons, and then Ti

becomes Ti4+

and O becomes O2-

. However, this is extremely oversimplified. Since both

ionic and covalent bonding characters coexist in TiO2, the valence states must be lower

than Ti4+

and O2-

. For pure TiO2, the calculated atomic charge density of Ti and O is

9.448e and 7.276e, which corresponds to Ti2.5+

and O1.25-

, respectively. In case of [001]

79

vacancy-ordered structure, while there is no change in atomic charge density on O atoms,

we find that the atomic charge density is increased in neighboring Ti atoms and that there

is a significant difference in charge density increase corresponding to the equatorial and

apical Ti atoms, 1.1 e/atom for equatorial Ti and 0.44 e/atom for apical Ti, respectively.

These results should be interpreted together with the partial density of states results in Fig.

5-4. As a consequence, the considerable increase of charge density in equatorial Ti atoms

-8 -6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Energy (eV)

-5

-4

-3

-2

-1

0

1

2

3

4

5

DO

S (

arb

. u

nit

s)

-8 -6 -4 -2 0 2 4 6-5

-4

-3

-2

-1

0

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Energy (eV)

-5

-4

-3

-2

-1

0

1

2

3

4

5

DO

S (

arb

. u

nit

s)

Vo VoVoVo Vo Vo

Vo

Vo

VoVo

Vo

Vo

VoVo

equatorialTi equatorial Ti

apical Ti apical Ti

equatorialTi

apical Ti

equatorialTi

apical Ti

(a) (b)

Fig. 5-4 Partial density of states of equatorial and apical Ti atoms surrounding oxygen

vacancies. (a) Vacancy-ordered structure along [001], and (b) vacancy-ordered

structure along [110].

80

leads to the enhanced metallic properties of these successive equatorial Ti atoms, and

then the connection of these Ti atoms by metallic bonds forms the conductive channel.

From the Bader charge analysis, in addition to atomic charge density, we can also

calculate the Bader volume of each atom. The calculated Bader volume of [001] vacancy-

ordered structure is hypothetically drawn in Fig. 5-5 assuming that all atoms are in

perfect spherical shape. Compared to Ti atoms that are far away from vacancies, the

volume of apical Ti atoms is increased by about 30%. On the other hand, the volume of

equatorial Ti atoms is increased by about 150%. In spite of the assumption of spherical

atom, equatorial atoms are already partially overlapped. If it is considered that electrons

are occupied by t2g orbital that is pointing at either [001] or [110], the larger part of Ti

V=17.2Å 3V=9.0Å 3 V=6.9Å 3

Fig. 5-5 Schematic illustration of atomic volume calculated by Bader analysis.

81

atoms will be overlapped. This suggests that there might be some charges that are shared

by two Ti atoms. Additionally, we find that most of the additional charges in equatorial

Ti (1.1 e/atom) atoms are distributed outside of the initial volume. Therefore these

increased charges can take part in the formation of metallic bonds between equatorial Ti

atoms along [001] direction.

Inspecting the iso-surface of band decomposed charged density of all defect states

within the band gap, as shown in Fig. 5-6, we show that a conductive channel is formed

in both [110] and [001] vacancy chain systems. This result suggests that the conduction

mechanism might be the defect assisted tunneling through metallic Ti atoms, not merely

electron doping by shallow donor defect level. Therefore, we conclude that the oxygen

vacancies may act as mediators of the electron conduction, whereas the actual electron

transport is conducted by successive Ti atoms in the channel. Based on these results, we

Ti

Ti

0.1e/Å 3

Ti

0.1e/Å 3

(a) (b)

Fig. 5-6 Iso-surface of partial charge density distribution of two vacancy-ordered

structures. The picture is drawn at 0.1e/A3 of iso-value. (a) Vacancy-ordered structure

along [110], and (b) vacancy-ordered structure along [001].

82

can suggest a more plausible theory about the “ON”-state conduction mechanism in rutile

TiO2.

5.3 STABILITY OF THE VACANCY-ORDERED STRUCTURE

We demonstrated that ordering of oxygen vacancies in certain direction forms the

filament type conductive path in rutile TiO2. In order to explain the “ON”-state

conduction, however, this vacancy-ordered structure must be energetically stable to

maintain the “ON”-state until additional bias, for instance voltage in opposite polarity in

case of bi-polar switching, is applied for switching to “OFF”-state. First, we calculate the

vacancy formation energy of some structures shown in Fig. 5-1 those have different

number of oxygen vacancies. The equation of the vacancy formation energy calculation

is as follows.

2.7

3.0

3.3

3.6

3.9 [110] [001]

846532

EV

F (

eV

/va

c.)

Number of vacancies1

Fig. 5-7 Vacancy formation energies with increasing number of oxygen vacancies

along [110] and [001] directions.

83

Evf E(TiO2x ) E(TiO2)n

2E(O2)

, (5-1)

where E(TiO2-x) is the total energy of a supercell containing oxygen vacancies and

E(TiO2) is the total energy of a perfect TiO2 in the same size of supercell. E(O2) is the

energy of the oxygen molecule and n is the number of oxygen vacancy. Fig. 5-7 displays

the calculated vacancy formation energy with increasing the number of oxygen vacancies

along two different directions, [110] and [001]. As the number of oxygen vacancies

increases, the energy required to form an oxygen vacancy is reduced. As a result,

vacancy-ordered structures in both directions show the lowest vacancy formation energy.

This means that these kinds of vacancy ordering are likely to be formed and retained as a

stable state in real TiO2.

0.30

0.32

0.34

0.36

0.38

0.40

FR3R2

Ev

f (eV

/fo

rmu

la u

nit

.)

Vacancy configurationR1

R1 : random config.

R2, R3 : random on (110)

F : Filament on (110)

Fig. 5-8 Vacancy formation energies of TiO2 supercell with different oxygen vacancy

configurations. Vacancies are randomly distributed in R1. In R2 and R3, all vacancies

are confined to the same (110) plane. F represents the vacancy-ordered structure in

[001] direction.

84

The stability of the vacancy-ordered structure has also been investigated by

comparing its formation energy with that of randomly distributed vacancy structures

using the same equation as above. For this calculation, the same number of oxygen

vacancies has been included in the supercell. All supercells contain 8 oxygen vacancies

which is the same number of oxygen vacancies as [001] vacancy-ordered structure has.

As shown in Fig. 5-8, the vacancy formation energy in [001] vacancy-ordered structure is

lower than other structures, implying that the vacancy ordering consisting of the closest

vacancies is the most favorable combination. Thus, once it is formed, the conductive

channel could be sustained even if bias is removed, which is critical for the operation of a

real device.

Meanwhile, it is expected that there should be a certain magnitude of an energy

barrier for the formation of vacancy-ordered structure. Among these random vacancy

configurations, the highest energy is observed for the R3 structure. This fact indicates the

existence of the barrier for oxygen vacancy movement to form the vacancy-ordered

structure.

5.4 ELECTRONIC BAND STRUCTURE

One method to describe the character of defect states is to plot energy band

structure. While only one or two k-points is included in the calculation of density of

states, energy band structure shows the shape of an individual energy level with respect

to many k-points. Therefore more detailed information about defect energy levels can be

obtained from the energy band diagram. We computed the electronic band structure of 3

x 3 x 4 supercell of rutile TiO2 which contains up to 8 oxygen vacancies.

85

First, the energy band diagram of TiO2 with an isolated single oxygen vacancy is

displayed in Fig. 5-9. Both spin-up and spin-down energy bands are presented. In

accordance with the density of states shown in Fig. 3-7, the defect energy state is

observed at about 0.4 eV below the conduction band minimum, which is regarded as a

deep defect level. Therefore electron doping at room temperature is not energetically

favorable. Furthermore the defect energy level is nearly flat, meaning that the electron in

this occupied defect state is strongly localized. According to the dispersion relations in

-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M

Spin up Spin down

Fig. 5-10 Energy band diagram of TiO2 supercell with di-vacancy.

-1

0

1

2

3

4

En

erg

y (

eV

)

ARZX M-1

0

1

2

3

4

E

nerg

y (

eV

)

ARZX M

Spin up Spin down

Fig. 5-9 Energy band diagram of TiO2 supercell with an isolated single oxygen vacancy.

86

wave propagation, the fact that the energy level is flat indicates that the electrons in this

energy level are not dispersive. In other words, the momentum and velocity of these

electrons are extremely small, thus it is not likely for these electrons to take part in the

conduction due to strong localization. Fig. 5-10 shows the energy band diagram of TiO2

with di-vacancy. The location of defect states is in a good agreement with the density of

states that is previously shown in Fig. 4-3(a). Similar to the single vacancy case, most of

the defect energy levels are almost flat.

On the other hand, the dispersion of defect energy levels becomes different if

-1

0

1

2

3

4

En

erg

y (

eV

)

ARZX M-1

0

1

2

3

4

En

erg

y (

eV

)

ARZX M

Spin up Spin down

(a)

-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M-1

0

1

2

3

4

En

erg

y (

eV

)

ARZX M

Spin up Spin down

(b)

Fig. 5-11 Energy band diagram of TiO2 supercell with ordered vacancies along (a)

[110] direction, and (b) [001] direction.

87

oxygen vacancies are ordered. It is found that the ordering of oxygen vacancy could

substantially increase the conductivity of TiO2 by changing the electronic configuration.

Fig. 5-11 presents the energy band diagram of two different vacancy-ordered structure,

[110] and [001], respectively. Unlike a single vacancy, defect energy levels have a slope

with respect to K-points, indicating that electrons in these energy levels are much less

localized than in the case of a single vacancy. Thus electrons become weakly bound or

delocalized and gain extra momentum. As a result, the resistivity of the system will be

significantly decreased.

In case of [001] vacancy-ordered structure, it should be noted that the defect

energy levels have a significant slope only along the line from Γ(000) to Z(001) that

corresponds to [001] direction. Fig. 5-12 represents high symmetry directions in the

tetragonal structure. The direction from Γ(000) to Z(001) is the same direction as the

vacancy chain. Therefore electrons have greater momentum especially in [001] direction

Fig. 5-12 Schematic diagram of the high-symmetry directions in the first brillouin

zone of the tetragonal structure.

88

than any other directions, and thus the substantial increase in the electrical conductivity

will be obtained along [001] direction. This fact verifies that the channel formed by

vacancy ordering is the metallic conductive channel which might be associated with the

“ON”-state in TiO2.

5.5 OXYGEN VACANCY ORDERING IN MAGNÉLI PHASE

Our results about the relevance of ordering of oxygen vacancies or metallic Ti

atoms in the conductivity enhancement have been emphasized by some previous works.

Similar calculation has been performed to investigate the effect of dislocation with

ordered vacancies on the conduction properties in SrTiO3 [1]. It argued the possible

Fig. 5-13 Phase diagram of the Ti-O system. TinO2n-1 is the so-called Magnéli phases.

89

electron transport in the direction of dislocation that is formed along the [001] direction.

More recently, it has been experimentally observed that the conductive filaments

are composed of Magnéli phase [2]. Rutile TiO2 exists in nonstoichiometric form,

therefore it is generally expressed by TiO2-x. If the sample is considerably reduced, that

corresponds to an increased x, i.e., the concentration of oxygen vacancies is increased

and long range order of oxygen vacancies is formed. This is so-called Magnéli phases [3-

4]. Fig. 5-13 shows the phase diagram of Ti-O system [5]. They form a homologous

series TinO2n-1 in which n is mostly 4 or 5. Compared to TiO2-x, Magnéli phases show

marked difference in their crystalline structure as well as in their magnetic and electrical

properties. It is known that Magnéli phase shows metallic conducting behavior at room

temperature [6-7].

We built the unitcell of Ti4O7 and calculated the partial charge density distribution

Fig. 5-14 Iso-surface of partial charge density distribution of a Ti4O7 unitcell. The

blue and red balls represent Ti and O respectively. Excess charges are found in every

Ti atoms.

90

as shown in Fig. 5-14. It is noted that excess charges are observed in all Ti atoms,

indicating that excess electrons are completely delocalized. Thus these electrons easily

contribute to the metallic conduction of Ti4O7 phase. Although the mechanism is not

understood well, long range order of bipolaron chain is somehow associated with the

metallic conduction in Magnéli phase.

In Magnéli phase, due to high concentration of oxygen vacancies, excess

electrons of more than 1022

/cm3 are induced, and these electrons cause lattice distortion

resulting in polaron that is a combination of electron and lattice distortion. Then, a pair of

polaron aligns in a certain direction, therefore long range order of bipolaron chain forms.

Fig. 5-15 presents the crystal structure of Magnéli (Ti4O7) phase that is oriented along the

direction of bipolaron chains, which is shown inside the dotted circle. It is composed of

Ti3+

Fig. 5-15 Schematic illustration of the crystal structure of the Magnéli phase (Ti4O7).

The blue and red ball represents Ti and O, respectively, and the yellow ball represents

Ti3+

. Two rows of oxygen atoms are missing in the bipolaron chain which is in the

dotted black circle.

91

two rows of successive Ti3+

atoms in one direction. This means that two oxygen vacancy

chains exist inside bipolaron chains. This vacancy chains with long range ordered

bipolarons are very similar to our simulated vacancy-ordered structure. Therefore it can

be concluded that vacancy ordering is strongly related to the formation of conductive

channel in rutile TiO2, which accounts for “ON”-state conduction.

These results suggest that the concentration of the oxygen vacancies might be an

important factor in the formation of conductive channel. If we assume that oxygen

vacancy concentration in rutile TiO2 further increases for some reason, rutile will be

transformed into Magnéli phase. This fact implies that there might be a critical vacancy

concentration which determines the specific geometry of the oxygen vacancies

distribution. In real ReRAM device, during electroforming or resistance switching,

oxygen vacancies move and additional vacancies are created by applied bias. This will

affect the concentration of oxygen vacancies. Thus if vacancy concentration exceeds a

critical point in a local area, vacancies start ordering to lower the total energy of the

system. Further study is needed to understand the effect of vacancy concentration.

92

REFERENCES

[1] K. Szot, W. Speier, G. Bihlmayer, and R. Waser, “Switching the electrical resistance

of individual dislocations in single-crystalline SrTiO3,” Nat. Mater., vol. 5, pp. 312-

320, Apr 2006

[2] D.-H. Kwon, K. M. Kim, J. H. Jang, J. M. Jeon, M. H. Lee, G. H. Kim, X.-S. Li, G.-

S. Park, B. Lee, S. Han, M. Kim, and C.S. Hwang, “Atomic structure of conducting

nanofilaments in TiO2 resistive switching memory,” Nat. Nanotech., vol. 5, pp. 148-

153, Feb 2010

[3] L. A. Bursill, and B. G. Hyde, “Crystallographic shear in the higher titanium oxides:

Structure, texture, mechanisms and thermodynamics,” Prog. Solid State Chem. Vol.

7, pp. 177, 1972

[4] M. Marezio, D. B. Mcwhan, P. D. Dernier, and J. P. Remeika, “Structural aspects of

the metal-insulator transitions in Ti4O7,” J. Solid State Chem., vol. 6, pp. 213-221,

Apr 1972

[5] G. V. Samsonov, “The oxide handbook,” IFI/Plenum Press, New York, 1982

[6] R. F. Bartholomew, and D. R. Frankl, “Electrical properties of some Titanium

Oxides,” Phy. Rev., vol. 187, pp. 828-833, Nov 1969

[7] A. D. Inglis, Y. L. Page, P. Strobel, and C. M. Hurd, “Electrical conductance of

crystalline TinO2n-1 for n=4-9,” J. Phys. C, vol. 16, pp. 317, 1983

93

[8] D. D. Cuong, B. Lee, K. M. Choi, H.-S. Ahn, S. Han, and J. Lee, “Oxygen vacancy

clustering and electron localization in oxygen-deficient SrTiO3: LDA+U study,”

Phys. Rev. Lett., vol. 98, pp. 115503, Mar 2007

94

CHAPTER 6: TRANSITION FROM “ON”-STATE TO “OFF”-

STATE

This chapter discusses the resistance switching from “ON”-state to “OFF”-state. The

effect of charge state of oxygen vacancies on the formation of conductive channel is

presented first. Then, the rupture of the conductive filament is discussed in detail. For this

purpose, the diffusion of oxygen atoms into the conductive channel is investigated. In

addition, the interaction between oxygen vacancies and hydrogen atoms is discussed. Thi

s is followed by a discussion for the effect of hydrogen atoms on the degradation of

“ON”-state conductivity. The chapter concludes with a proposed resistance switching

mechanism based on our results.

95

6.1 RUPTURE OF THE CONDUCTIVE CHANNEL

The resistance switching of many binary transition oxides such as TiO2, NiO2,

Nb2O5 is known as a fuse-antifuse type switching. In this category, the dominant

resistance switching mode is unipolar switching. In this type of resistance switching,

thermal effects have a major impact on the reset switching. The reset switching can be

attributed to local power dissipation due to high current density in the conductive path.

This power dissipation induces high temperature in the conductive path causing either the

diffusion of oxygen vacancies out of the conductive channel or the diffusion of other

impurities into the conductive channel, or a phase transition may take place. Therefore

locally ruptured part of the conductive channel becomes insulating so that overall

resistance of the channel increases.

(a) (b)

Fig. 6-1 Schematic diagrams of (a) typical unipolar resistive switching behavior, and

(b) electroforming, set and reset switching process. Initial state (as-prepared sample),

and (1) forming, (2) reset, and (3) set processes.

96

Several papers have reported that the rupture of conductive channel occurs at the

anode interface [1-4]. Fig. 6-1 shows a possible driving mechanism for filament-type

resistive switching that shows unipolar switching behavior [5]. During the forming

process, filamentary conducting paths form as a soft breakdown in the dielectric material.

Then rupture of the conductive filaments takes place at the interface. Thermal redox

and/or anodization near the interface between the metal electrode and the oxide is widely

accepted as the mechanism behind the formation and rupture of the conductive filaments.

On the other hand, it is also reported that rupture of the conduction path can occur in the

middle of the conduction path rather than the anode side [6]. The detail of the reset

switching is still obscure.

6.2 DIFFUSION OF OXYGEN VACANCY

For the reset switching, it can be assumed that oxygen vacancies diffuse out of the

conductive channel by thermal joule heating. In other words, oxygen atoms from outside

Fig. 6-2. Schematic diagram of vacancy ordered structure containing one oxygen

atom in the channel. One oxygen vacancy is exchanged with an oxygen atom

adjacent to the channel.

97

diffuse into the channel. Fig 6-2 shows the schematic diagram of vacancy-ordered

structure containing one oxygen atom in the middle of the vacancy channel. One oxygen

vacancy in the channel is exchanged with an oxygen atom adjacent to the vacancy

channel. The energy band diagram of this structure is presented in Fig. 6-3. For

comparison, the energy band diagram of vacancy-ordered structure is also displayed. The

result shows that the diffusion of one oxygen vacancy out of the vacancy channel makes

most of the defect energy levels flattening. Therefore electrons become strongly localized

in these defect energy levels. This suggests that the conductivity of this system is

substantially decreased by reducing the momentum of the defect states electrons.

-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M-1

0

1

2

3

4

En

erg

y (

eV

)

ARZX M

Spin up Spin down

(a)

-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M

Spin up Spin down

(b)

Fig. 6-3 Energy band diagram of (a) vacancy-ordered structure, and (b) vacancy-

ordered structure in which one oxygen vacancy diffuses out of the channel.

98

Fig. 6-4 shows the schematic diagram of vacancy-ordered structure in which two

oxygen vacancies diffuse out. We calculated the total density of states as shown in Fig. 6-

5. It seems that the total density of states is very similar to that shown in Fig. 6-3, in

which one oxygen atom is incorporated. Flat defect energy levels will freeze all electrons

in defect states by strongly localizing them in Ti 3d orbitals. This will significantly

increase the resistance of this vacancy chain, resulting in an insulating channel. Therefore

the diffusion of oxygen atom might be the major factor to lead the resistance switching

from “ON”-state to “OFF”-state.

In order to have a better understanding, the calculation of partial charge density

distribution was carried out, as shown in Fig. 6-6. Due to two oxygen atoms in the middle

of channel, the conductive path is completely disconnected. That is, metallic bonds

between Ti atoms in the vertical direction are broken by two oxygen atoms. This means

that the propagation of waves of each energy level is interfered; hence the transport of

electrons across these oxygen atoms is not likely to occur. This result is in good

agreement with the energy band structure.

Fig. 6-4 Schematic diagram of the vacancy-ordered structure containing two

oxygen atoms in the channel.

99

-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M-1

0

1

2

3

4

ARZX

En

erg

y (

eV

)

M

Spin up Spin down

Fig. 6-5 Energy band diagram of the vacancy-ordered structure in which two oxygen

vacancies diffuse out of the channel.

Ti

O

Vo

0.1e/Å 3

Fig. 6-6 Iso-surface of partial charge density of the vacancy-ordered structure in which

two oxygen vacancies diffuse out of the channel.

100

6.3 EFFECT OF HYDROGEN IMPURITIES

It is known that not only intrinsic defects but also extrinsic impurities such as Cr,

Co, and Ni affect the resistance switching in many resistance switching materials [7-9].

These impurities can act as dopants that are of critical importance in semiconductor

devices. Small amounts of dopants that act as donors or acceptors are introduced into the

semiconductor crystal lattice to induce a significant change in the electronic properties of

the semiconductor.

Among those impurities, it is known that hydrogen also can affect the chemical

and electronic properties of TiO2 [10-11]. Due to its small size and high reactivity,

hydrogen diffuses in the lattice and strongly interacts with host atoms, impurities, and

defects. As an interstitial impurity, hydrogen tends to lose its electrons by behaving as a

donor and forming OH- groups. Then, the excess electrons are transferred to the cations,

thus reducing Ti4+

to Ti3+

. In such case, hydrogen is expected to behave as a shallow

donor dopant, which can induce an energy level close to the conduction band [12-13].

On the other hand, hydrogen can neutralize the effect of native defects such as

vacancies, by saturating the corresponding dangling bonds [14-15]. In ZnO, it has been

theoretically reported that hydrogen forms a complex with an oxygen vacancy, where it

takes the place of the missing oxygen atom, and binds covalently to the surrounding Zn

atoms [16]. The formation of such a hydrogen-vacancy complex has important

consequences, because it can change the deep donor character of the oxygen vacancy,

thus resistance switching character of TiO2 could be affected by hydrogen atoms. J. R.

Jameson has reported that hydrogen ions are the mobile dopant responsible for the field-

programmable rectification observed naturally in the Pt/TiO2/Pt system [17]. They

101

speculated that the motion of hydrogen might also be a previously unrecognized source

of resistive switching in Pt/TiO2/Pt and related devices. However the interaction of

hydrogen with TiO2 has been much less intensively investigated than the effect of oxygen

vacancies.

6.3.1 INTERSTITIAL HYDROGEN

First, the effect of an interstitial hydrogen atom has been investigated. We

inserted a hydrogen atom in the middle of open channel in perfect TiO2, as shown in Fig.

6-7. After relaxation, hydrogen binds to oxygen as H+ ion by giving an electron to Ti

atom. In order to find hydrogen induced defect states, we calculated the total density of

states. As is shown in Fig. 6-8, interstitial hydrogen creates a defect energy level that is

very close to the conduction band, which makes hydrogen act as an n-type dopant. Note

that only positive spin energy level is observed. This is due to one excess electron

donated by one hydrogen atom. This unintentional doping by hydrogen atoms may cause

unwanted increase in “OFF”-state leakage current.

H

O

Ti

Fig. 6-7 Schematic diagram of one interstitial hydrogen in the perfect TiO2 supercell

102

-8 -6 -4 -2 0 2 4 6-200

-150

-100

-50

0

50

100

150

200

DO

S (

arb

. u

nit

s)

E-EF (eV)

Fig. 6-8 Total density of states of TiO2 with one interstitial hydrogen atom.

H

Vo H + Vo

Before Relaxation After Relaxation

(a) (b)

Fig. 6-9 Schematics of TiO2 supercell with one oxygen vacancy and one hydrogen

atom. (a) Before relaxation, and (b) after relaxation.

103

6.3.2 HYDROGEN-VACANCY COMPLEXES

As discussed above, hydrogen can make the complex with oxygen vacancy. Fig.

6-9 shows the schematic diagram of TiO2 supercell with one oxygen vacancy and one

hydrogen atom. During relaxation, hydrogen that was placed in the middle of open

channel moves to vacancy site, then take the place of vacancy. This implies that oxygen

vacancy acts as a stable site for a hydrogen atom. We found that this hydrogen-vacancy

complex structure is about 0.46 eV lower in energy than isolated hydrogen and vacancy.

Therefore hydrogen atoms will make the complex with oxygen vacancies if there is

enough energy to overcome the barrier for hydrogen migration.

The electronic structure of hydrogen-vacancy complex is presented in the total

density of states as illustrated in Fig. 6-10, which is characterized by a defect energy level

-8 -6 -4 -2 0 2 4 6-200

-150

-100

-50

0

50

100

150

200

DO

S (

arb

. u

nit

s)

E-EF (eV)

0.2 eV below CBM

Fig. 6-10 Total density of states of TiO2 with hydrogen-vacancy complex.

104

located at 0.2 eV below the conduction band minimum. It should be noted that there is

only one unpaired singly occupied defect state in the band gap. Since there are two

defects, hydrogen and vacancy, three defect states are expected to be presented in the

band gap. In a simplified picture, this indicates that two out of three excess electrons

contribute to pile up electronic charge on hydrogen-vacancy complex and are

accommodated in energy levels deep in the valence band. In other words, one out of two

defect states which are induced by an oxygen vacancy can be eliminated from the band

gap by formation of hydrogen-vacancy complex. Therefore it is quite necessary to

consider the impact of hydrogen atoms on the conductive channel which is associated

with oxygen vacancy ordering.

6.3.3 RESET SWITCHING BY HYDROGEN

In order to examine the interaction between hydrogen and vacancy-ordered

conductive channel, we assumed that all vacancies in the conductive channel are

occupied by hydrogen atoms as demonstrated in Fig. 6-11. The total density of states is

displayed as well. In comparison to vacancy-ordered structure, the number of defect

states is considerably reduced by adding hydrogen atoms in vacancy sites. Therefore

defect assisted tunneling from valence band to conduction band might be interrupted, and

then the resistance of this system will be increased.

This is due to the strong electron localization in hydrogen atoms. Fig. 6-12 shows

the electron localization function and iso-surface of partial charge density distribution of

hydrogen-vacancy complex chain structure. Note that strong electron localization is

found in the place of combined hydrogen-vacancy. This result indicates that electrons

that were in the removed defect states from the band gap are localized in hydrogen atoms,

105

and thus hydrogen reduces the charge density of Ti atoms. Although some electrons

remain in t2g orbital of Ti 3d, metallic network between Ti atoms in vertical direction are

disconnected. The bond breaking between Ti atoms is well illustrated in iso-surface of

partial charge density distribution. Therefore the diffusion of hydrogen into conductive

channel and formation of complex with oxygen vacancy can induce the rupture of

conductive path, and then it results in the transition from “ON”-state to “OFF”-state by

increasing the resistance of the vacancy channel.

Vo chain Vo + H chain

Ti

O

Vo

Vo+H

(a)

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

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

E - EF (eV)

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

-100

-50

0

50

100

150

DO

S (

arb

. u

nit

s)

E - EF (eV)

(b)

Fig. 6-11 (a) Schematics of the vacancy chain structure and hydrogen-vacancy complex

chain structure. (b) Total density of states.

106

To investigate the modification of charge density, Bader charge analysis was

carried out. In comparison to vacancy-ordered structure, after the addition of hydrogen

atoms, the charge density of hydrogen is increased by 0.6e, while the charge density of

equatorial Ti in the conductive channel is decreased by 0.48e. Localized charges in

hydrogen-vacancy complex site are not likely to contribute electron conduction since

those are occupying energy levels in deep valence band. Furthermore equatorial Ti atoms

that are a part of the conduction path are deprived of metallic properties by loosing

charges, and then they become more ionic. As a result, metallic bond will be broken and

resistance of this channel will be increased, which might be regarded as a part of the reset

switching.

6.4 PROPOSED RESISTANCE SWITCHING MODEL

Based on our results, we propose the overall resistance switching model. The

schematic diagram of the switching model is illustrated in Fig. 6-13. At the initial state,

0.1e/Å 3

Ti

Vo

H

Ti

H+Vo

O

(a) (b)

Fig. 6-12 (a) Electron localization function of hydrogen-vacancy complex chain. (b)

Iso-surface of partial charge density distribution of hydrogen-vacancy complex chain.

107

as-deposited TiO2 has some amount of oxygen vacancies as native defects. The

concentration of oxygen vacancies could vary depending on the deposition process.

While electroforming, due to applied bias, oxygen vacancies migrate into the cathode,

pile up there and further grow toward the anode. During this process, considerable

amount of additional oxygen vacancies could be generated by migration of oxygen

toward the anode interface. This migration of oxygen leaves oxygen vacancies behind in

the bulk. A few literatures have reported that the evolution of oxygen gas at the anode

might be attributed to the formation of oxygen vacancies in TiO2 [18-19]. Therefore

oxygen vacancy concentration will be substantially increased in local area, in which

vacancies start self-ordering.

Fig. 6-13 (b) represents the schematics of “ON”-state which corresponds to the

low resistance state. A conical shape of filament type conductive path is formed, which is

(a) (b) (c)

Fig. 6-13 Schematic illustration of resistance switching model. (a) Initial as-sampled

state. Oxygen vacancies are randomly distributed. (b) “ON”-state (Low Resistance

State), and (c) “OFF”-state (High Resistance State).

108

composed of many vacancy-ordered domains. The overall resistance of this state might

be determined by the number or size of these domains. During reset switching, these

vacancy-ordered domains are destroyed by thermal effect. This can occur by the diffusion

of either oxygen or hydrogen atoms into vacancy-ordered domains, resulting in high

resistance state.

109

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112

CHAPTER 7: CONCLUSION

In this dissertation, the mechanisms of “ON”-state conduction and resistance switching in

rutile TiO2 were elucidated through first-principle study using density function theory.

This chapter summarizes the contributions of this dissertation, consisting of the electronic

effect of oxygen vacancy and in-depth understanding of “ON”-state conduction and

resistance switching mechanism. Based on these contributions, future research works are

suggested.

113

7.1 CONTRIBUTIONS

First, electronic correlation effect in reduced rutile TiO2 was characterized by

LDA method including on-site Coulomb corrections between 3d orbital electrons of Ti

atoms and 2p orbital electrons of O atoms. Although there have been many studies

involving oxygen vacancy in TiO2, the effect of Coulomb interaction between 2p orbital

electrons has never been reported. We found that the addition of on-site Coulomb

corrections for 2p orbital electrons provides an improved description of the electronic

properties of rutile TiO2. The obtained physical properties and the band gap energy is in

very good agreement with experimental values.

One of the main achievements of this dissertation was the understanding of the

role of oxygen vacancies in the electronic structure in TiO2. Strong electron localization

and deep level of defect states of an isolated single oxygen vacancy indicate that the

effect of vacancy on the conductivity is almost negligible. The effect of di-vacancy was

also investigated. The most stable configuration of two oxygen vacancies is equatorial di-

vacancy in which two vacancies are in the nearest distance. The strong interaction

between two vacancies induces the splitting of defect states in the band gap. Most of the

excess charges in these defect states are localized in equatorial Ti atoms that are

positioned between two vacancies. These excess electrons that are occupied by t2g orbital

in Ti increase the charge density of Ti, and thus Ti becomes more metallic. These

equatorial Ti atoms form t2g-t2g type metallic which is expected to increase the

conductivity of TiO2.

The most valuable achievement of this work was the fundamental understanding

of “ON”-state conduction mechanism in TiO2. From the investigation of many differently

114

configured vacancy clusters, it was found that vacancy-ordered structure shows the

lowest vacancy formation energy. Some important evidence revealing the role of vacancy

ordering in the formation of “ON”-state was obtained from the calculation of electronic

properties, such as electron localization function, electronic band structure, and partial

charge density distribution. In contrast to single or di-vacancy, the energy band diagram

of vacancy-ordered structure shows that electrons in defect states are weakly localized,

therefore those electrons are allowed to have some momentum to drift under the applied

bias. Furthermore, the formation of the conductive channel was visualized by plotting

iso-surface of partial charge density distribution, showing the formation of metallic bonds

between Ti atoms. Therefore, the resistance of the system must be significantly decreased

by vacancy ordering. Based on these results, it can be suggested that vacancy ordering is

deeply associated with “ON”-state conduction in rutile TiO2.

Finally, another unique accomplishment of this dissertation was that theoretical

investigations of reset switching were performed by considering the diffusion of oxygen

and hydrogen. Energy band diagram shows that the momentum of electrons in defect

states is substantially decreased by the diffusion of oxygen atom into the conductive

channel. Added oxygen atom in the channel breaks a metallic bond between Ti, and then

disconnects the conductive channel. The diffusion of hydrogen atoms into the channel

might also be responsible for the reset switching. Hydrogen combined with oxygen

vacancy reduces the charge density of adjacent Ti atoms by localizing those charges in

the deep valence band, and thus Ti atoms become more ionic. Hence the resistance of the

channel is further increased.

115

7.2 FUTURE WORKS

All theoretical calculations in this dissertation were performed at 0K. In order to

have a more realistic understanding, it is quite necessary to take into account thermal

effects on the resistance switching. Temperature will significantly affect the dynamic

behavior of oxygen vacancies that has great importance as a part of the resistance

switching process. In addition, understanding of the thermal effects would be critical to

consider the retention or endurance properties. Therefore the study of dynamics of

oxygen vacancies would give us more in-depth understanding of the resistance switching.

AIMD (Ab-initio Molecular Dynamics) can be suggested as one of the approaches to

simulate the dynamics of TiO2 during resistance switching.

In this dissertation, we have investigated the behavior of oxygen vacancies in bulk

rutile TiO2. In real devices, however, switching material is sandwiched by two electrodes

on both sides. Therefore the integration of MIM (metal/insulator/metal) system will be

required. Thus more careful works must be focused on the interface between metal and

switching material to understand the effect of interfaces on the resistance switching. In

addition, electron transport calculations should be carried out based on Non-Equilibrium

Green‟s Function formalism (NEGF) to determine the I-V characteristics for. both “ON”-

state TiO2 and “OFF”-state TiO2, as proposed by this dissertation. We believe that this

will give more improved in-depth understanding of switching mechanism which may

provide guidance for the direction of development of future ReRAM devices.