Two dimensional naxsis as a promising anode material for rechargeable sodium based batteries ab initio material design

vi LIST OF ABBREVIATIONS 2D: Two-dimensional DFT: Density functional theory DOS: Density of State EES: Electrical energy storage GGA: Generalized Gradient Approximation LDA: Local densi

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY -

PHAM THI DUNG

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY -

PHAM THI DUNG

i

ACKNOWLEDGMENT

I am truly honored to submit my master thesis for the degree of Master at Nanotechnology Program, Vietnam Japan University This work has been carried out in the Nanotechnology program, Vietnam Japan University, Vietnam National University of Hanoi

I would like to express my sincere thạnks to my supervisor: Dr Dinh Van An, JICA expert, lecturer, Vietnam Japan University (VJU), Vietnam National University (VNU) for accepting me as his student, for guidance, and his encouragement to complete this research

I would like to thank all students and teachers of Nanotechnology, Vietnam Japan University, Vietnam National University of Hanoi for the pleasant and stimulating atmosphere during my research study

Hanoi, May 25, 2018

Student

Pham Thi Dung

ii

TABLE OF CONTENTS

ACKNOWLEDGMENT i

TABLE OF CONTENTS ii

LIST OF FIGURES iv

LIST OF TABLES v

LIST OF ABBREVIATIONS vi

INTRODUCTION 1

CHAPTER 1 LITERATURE REVIEW 3

1.1 Energy demand 3

1.2 Rechargeable batteries 3

1.3 Electrode materials 6

1.3.1 Cathode material 7

1.3.2 Anode material 7

1.3.2.1 Silicene, germane, stanene, phosphorene and borophene 8

1.3.2.2 2D transition metal oxides 9

1.3.2.3 2D transition metal dichalcogenides 10

1.3.2.4 2D transition metal carbides/nitrides 11

1.3.2.5 Emerging 2D materials 11

1.4 Purpose of thesis 12

CHAPTER 2 METHOD 13

2.1 Density functional theory 13

2.1.1 Hohenberg - Kohn theorems 15

2.1.2 Exchange-Correlation Functionals 17

2.1.3 Solving Kohn-Sham Equation 18

2.2 Nudged Elastic Band method 18

2.3 Calculation Scheme 20

CHAPTER 3 RESULTS AND DISCUSSION 21

3.1 Atomic structure of NaxSiS (x= 0-0.5) 21

3.1.1 Host material SiS (x=0) 21

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3.1.2 Na1Si18S18 (x=0.05) 22

3.1.3 Higher Na concentration NaxSiS (x= 0.111-0.5) 24

3.2 Theoretical capacity and open circuit voltage 28

3.2.1 Theoretical capacity 28

3.2.2 Open circuit voltage 28

3.3 Adsorption energy 29

3.4 Electronic structure 31

3.5 Diffusion mechanism of sodium on the surface of NaxSiS 36

CONCLUSION 39

REFERENCES 40

iv

LIST OF FIGURES

Page

Figure 1.1 Schematic operation of batteries 5

Figure 2.1 An illustration of the self-consistent field 18

Figure 2.2 Decomposition of force in an image 19

Figure 3.1 The optimized structure of silicene sulfide The blue and yellow balls represent for Silicon and Sulfur atoms, respectively 21

Figure 3.2 Adsorption area of Na 23

Figure 3.3 Symmetrical adsorption sites of Na ion 24

Figure 3.4 Possible adsorption sites of the second Na ion 25

Figure 3.5 Possible configurations of Na0.167SiS 26

Figure 3.6 The most stable structures of different sodium concentrations (x=0.05-0.5) 27

Figure 3.7 The adsorption energy diagram 30

Figure 3.8 The density of state of pristine SiS Fermi level is set at zero 32

Figure 3.9 The density of state of Na0.05SiS Fermi level is set at zero 33

Figure 3.10 The density of state of Na0.111SiS Fermi level is set at zero 34

Figure 3.11 The density of state of Na0.278SiS Fermi level is set at zero 35

Figure 3.12 Diffusion pathway of sodium along a-direction 37

Figure 3.13 Diffusion pathways of sodium along b-direction 37

Figure 3.14 The diffusion pathway of single Na atom on SiS surface 38

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LIST OF TABLES

Page

Table 1.1 Sodium and Lithium characteristics 6

Table 3.1 The optimized structure parameter of silicene sulfide 22

Table 3.3 The sodium adsorption energies (Adsorption energy is in units of eV per

number of Na atoms) 29

vi

LIST OF ABBREVIATIONS

2D: Two-dimensional

DFT: Density functional theory

DOS: Density of State

EES: Electrical energy storage

GGA: Generalized Gradient Approximation

LDA: Local density approximation

LIBs: Lithium-ion batteries

OCV: Open circuit voltage

PAW: Projector augmented-wave

SES: Stationary energy storage

SIBs: Sodium ion battery

VASP: Vienna ab initio Simulation Package

NEB: Nudged Elastic Band method

1

INTRODUCTION

Nowadays, energy security is the goal of many countries, including Vietnam The shortage of energy, the effective use of recycling energy, and environmental pollution are emergency problems that demand strong efforts of scientists and governments In general, energy comes from two sources: nonrenewable source (fossil carbon such as oil, coal, and gas) and renewable source (wind, water, sun) The excessive exploitation of fossil energy leads to serious environmental problems such as pollution, global warming, and ozone depletion Therefore, the use of renewable energy is promoted to study and apply in the world, this essential solution is likely to deal with the lack of energy issue However, the efficient storage of renewable energy is the big challenge because almost renewable resources are intermittent sources Therefore, the study of efficient energy storage devices is extremely necessary The various electricity storage devices were designed such as batteries, capacitors, and the other devices Unlike primary batteries, secondary batteries are more popular for their charge and discharge abilities in use

The performance of rechargeable batteries is determined by theoretical capacity, voltage, electric conductivity, and life cycle parameters, strongly depends

on electrodes and electrolyte materials Therefore, the investigations to find out suitable materials for batteries have attracted much attention by both experiment and computational science

Today, thanks to the improvement in computational technology, researchers are making vigorous progress in the understanding of materials at atomic and molecular levels With this understanding, we can select suitable materials for specific purposes and also improve advanced materials for applications Aiming at enhancing the collaboration between experimental research and simulation work in studying on existing, and new materials as well as their application, computational material science with its techniques is applied to solve material relating problems

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Moreover, computational experiments have an advantage over real experiments because most of the variables can be controlled in experiment processes Nanotechnology may gradually take the forefront of scientists in the computational material science due to their nanoscale for several decades At electronic level, density functional theory (DFT) is a popular method to investigate material characteristics in quantum physics

Regarding the requirement of stable storage electric devices and the demand

of understanding material properties, we have investigated the performance of dimensional (2D) material as a promising anode for rechargeable batteries by simulation methods In this research, electronic properties and stable structures of

two-new materials Na-silicene sulfide were systematically investigated as a promising

anode for rechargeable sodium-ion batteries

3

CHAPTER 1 LITERATURE REVIEW

1.1 Energy demand

In recent decades, due to the growth of population and industry the rapid rising

of energy demands is an emerging issue A nearly double increase in total energy consumption from 549 quadrillion Btu to 815 quadrillion Btu over a period of 32 years, starting in 2012, has been predicted [1] To meet the increasing energy demand and respond to environmental impacts, renewable energy has been supported to expand use Nonetheless, most of renewable energies are from intermittent sources, which can only be used effectively if they are stored in sustainable electrical energy storage (EES) systems [2] Among numerous types of electricity storage, in recent years, rechargeable batteries have been intensively

studied and received most attention Basing on the estimation of International Renewable Energy Agency (IRENA) in 2015, there will be a rapid rise in the total

electricity stored in EES from just 0.8 GW in 2014 to 250 GW in 2030 [3] As one type of storage devices, rechargeable batteries have been studied and used as innovative devices with a wide application range of portable electronics, electric vehicles, power tools and stationary energy storage (SES) systems

1.2 Rechargeable batteries

Batteries are devices that convert the chemical energy into electric energy through an electrochemical redox reaction Batteries are classified generally into primary (nonrechargeable) and secondary (rechargeable), depending on their

rechargeability A battery often consists of one or more basically electrochemical

units (cell), connected in series or parallel, or both, depending on the requirement of voltage and capacity.[4]

In contrast to the secondary battery, the primary battery (or cell) is designed

to discharge once and is then discarded The typical primary cell is a zinc-carbon dry cell, which creates electric current from a redox reaction between zinc and manganese dioxide The primary battery is convenient with its low cost, and

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lightweight in the use for portable and electronic devices such as cameras, toys, and memory backup However, the primary cell is considered as an unfriendly environmental device due to the containing toxic heavy metals and strong acids The notable feature of the secondary battery (or rechargeable battery) is the reversible charge and discharge processes Rechargeable lithium-ion batteries (LIBs) is the most common battery, which has successfully applied for portable devices, electric vehicles since their first commercialization in the early 1990s[5, 6] Like all batteries, LIBs consists of three main components, including:

The anode or negative electrode, which provides electrons to the external

circuit in discharging process – is oxidized during the electrochemical reaction Since its commercialization in the 1990s, the carbon anode in LIBs has been chosen

as an effective electrode [7] for its low cost, large abundance, high electrical conductivity, and stable during charge/discharge process

The cathode or positive electrode, which receives electrons from external

circuit in discharging process – is reduced during the electrochemical reaction LiCoO2 was firstly investigated by Goodenough [8] and was originally commercialized by SONY With its high theoretical capacity of 155 , high voltage of 3.9 V, and good cycle performance LiCoO2 is currently used for commercial LIBs [9]

Electrolyte or separator is an ionic conductor – which gives the medium for

transfer ions, between anode and cathode The electrolyte can be in different forms

of ionic liquid, molten salt or solid [10]

In charge/discharge processes, the lithium intercalation and deintercalation can

be generally displayed by using the following equation:

[cathode] + (xj-xi) Li [anode] [cathode] + [anode]

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In charging process, lithium (metal) ions diffuse inside cathode then pass through the electrolyte and flow into the anode At the same time, electrons also transfer from cathode to anode in the external circuit After that, electrons and ions combine at the anode until fully charged

In discharging process, metal ions flow back through the electrolyte from anode to cathode Electrons flow from anode to cathode through the outer circuit, which is powering the electric devices Then the ions and electrons combine at the cathode When all the ions have migrated into the cathode, the battery finishes to discharges and needs charge again

Charging and discharging processes are displayed in Figure 1.1

Figure 1.1 Schematic operation of batteries [11]

Nowadays, the increasing demand of lithium and its small resource in the Earth result in the urgent problem that it is needed to find a more complementary battery material/source with high energy density and efficiency, long cycle and, low cost Sodium, which is the most popular alkali metal, is a sixth abundant element in the Earth crust Moreover, the chemical properties of Sodium are similar to Lithium because Sodium is the lithium’s next element in the periodic table Sodium batteries

6

(SIBs) become an ideal alternative to LIBs because of the pricing advantage and the similar batteries performance, compared to LIBs Corresponding to the almost infinite of sodium precursor deposit, the cost of sodium carbonate (about $135-$165 per ton) is much lower than lithium precursor lithium carbonate (about $5000 per ton in 2010) [5], which enhances the development for research of SIBs in large-scale batteries application Furthermore, the size of a lithium ion is much smaller than that of a sodium ion, requesting some specific difference in electrode materials, compared with LIBs, thereby promoting research about materials for electrodes in SIBs [12-14] Sodium and Lithium characteristics are presented in table 1.1

Table 1.1 Sodium and Lithium characteristics [15]

Category Lithium Sodium

In principle, the difference between cathode and anode in chemical potential is determined as working voltage (VOC) of batteries, which is calculated by the following formula:

(1.1)

(1.2) Where n is the number of charge carrier, and F, M indicate the Faraday constant and the molecular weight of the active material used in the electrode, respectively

Moreover, electronic and ionic conductivity play important roles in the intercalation/deintercalation of charges and ions within electrode materials In a battery, they are presented through determining the electronic properties and the activation energy (eV) of diffusion paths

1.3.1 Cathode material

Cathode materials determine voltage values of batteries, so most of these materials contain transition metals Their redox couples create a voltage in batteries such as NaFeO2 [16], NaCoO2 [17], Na3V2(PO4)3 [18] and NaMnO2 [19] The maximum energy density of SIBs can be reached by raising the working voltage of cathode or reducing the working potential of the anode and increasing electrode

capacity

1.3.2 Anode material

For anode, materials having high ion conductivity and large theoretical capacity were studied In contract to LIBs, graphite, the most popular anode materials, showed poor capacity into carbon host regarding SIBs [20] Various carbon-based materials, such as hard carbon [21], soft carbon [20], expanded graphite [22], carbon nanowires [23], carbon nanospheres [24] have investigated to develop the capacity of carbon-based materials However, it is hard to enhance their

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capacity due to the limited sodium host sites in carbon structures [25] In addition to carbon-based materials, metals, metal alloys, oxides were also investigated to be anode materials to extend the anode capacity and cycle life

Among various investigated anode materials, two dimensional (2D) materials which attract the wide attention of scientists for batteries by their specific characteristics such as high surface area and high concentrate of open-transport channels [26], help them to outperform their bulk counterparts From graphite, graphene was successfully isolated which was created a new concept: two-dimensional (2D) materials Nowadays, the beyond graphene or other 2D materials have been widely studied, including the compounds of transition metal like oxides (TMOs), dichalcogenides (TMDs), carbides/nitrides (Mxenes) and elemental analogues of graphene The scientific research were showed that the performance of those materials better than their bulk counterparts in various applications In comparison with their bulk, the distinctive properties of 2D materials due to the change their density of state which transform the way excitation energy interacts with the valence electron, so in 2D materials band gap is highly sensitive Moreover, in terms of morphology, their 2D materials have a high surface-to-mass ratio, providing a wide of active sites, corresponding to higher theoretical capacity

Their layers of 2D material are linked by weak van der Walls forces, which

contribute a negligible change of volume To sum up, 2D materials with their outstanding features, such as high capacity, stable structure, high conductivity in use, become ideal candidates for batteries materials Some 2D materials which have been studied for rechargeable batteries, are:

1.3.2.1 Silicene, germane, stanene, phosphorene and borophene

From the research on specific properties of graphene, other 2D material of IIIA, IVA, VA elements have been studied in both experimental and theoretical research The 2D structure of IVA elements (silicon, germanium, and tin) differ with graphene in monolayers, in detail, their structure like “low-buckled” (not

planar) The results of Ab initio proved the stable structure of those materials which

9

are suitable for batteries materials Silicene, germanene, and stanene were successfully fabricated by CVD method, and their structures were investigated by simulation methods These materials have been explored as anode materials for akaline-ion batteries with capacity of 954 mAh/g (silicene), 369 mAh/g (germanene), and 226 (stanene) [27-29]

Phosphorous is stable in numerous structures like white, red, black and violet allotropes Among which black phosphorous is the most stable structure It was studied to apply as anode material for Li/Na-ion batteries Nevertheless, the lattice structure of black phosphorous can be changed at high ion concentration in lithium intercalation process [30] From DFT calculations, phosphorene has strong binding with lithium resulting in capacity of 432.79 and low activation energy for diffusion of Li (0.08-0.68 ) [31] Additionally, blue phophorene has also been expected as a potential material for Li-ion batteries [32] Moreover, experiments have been proved the extra-large capacity for Na battery of phosphorene/graphene hybrid [33]

Another promising anode material in IIIA group is borophene [34] Jiang et al

used DFT calculation to demonstrate potential properties of boron, namely high theoretical specific capacity, large adsorption energy and low barrier energy Specifically, 1860 for lithium and 1218 for sodium are theoretical specific capacity of borophene, which are about 4 times higher than that of the graphite anode (372 ) The barrier energy along furrow of corrugated borophene is 2.6 with Li and 1.9 for Na

1.3.2.2 2D transition metal oxides

Transition metal oxides (TMOs) are known as popular compounds in nature, which have cation-exchange capability Electronic properties of 2D TMOs materials were studied with a wide range from metals to insulators, which have been found for the application of energy storage TMOs provide potential characteristics for electrodes like high capacity, stability and low cost Moreover, 2D transition Metal

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Hydroxides have been also studied for anode material The theoretical capacity of NiO is calculate to be 718 [37] The NiO nanosheet electrode has the capacity of 851 at the end of the 170th cycle at 2 [35] 2D Ni(OH)2 was

successfully synthesized via microwave under low temperature [36] The capacity

of Ni(OH)2 was calculate as 1874 F/g at 12 A/g [37] Ni(OH)2 nanoplates grown on graphene showed the good capacity and cycling performance [37] Co3O4 has long been studied for LIBs with the theoretical capacity of 890mAh/g corresponding 8 Li/unit [38] The porous Co3O4 nanosheets has specific capacity as well as 1450 remaining after 25 cycles [39] Co(OH)2 which has two polymorphs with hexagonal layers structure, has been developed as a potential material for supercapacitors (SC) [40]

1.3.2.3 2D transition metal dichalcogenides

Transition metal dichalcogenides (TMDs) are generally represented as MX2

formula In which, M is a transition metal from IV to X group and X is a chalcogen elements (Se, S and Te) Metal atoms are sandwiched between two layers of chalcogen atoms by covalent bond 2D TMDs were synthesized by CVD method from their bulk structure Electronic properties of 2D TMDs are ranged from metal (NbS2, VSe2) to insulator (HfS2) In the group of TMDs, MoS2, SnS2, and VS2 were studied as promising anode materials for LIBs and SIBs

NbSe2 monolayer was explored as anode 2D material for both of SIBs and LIBs The results showed the higher performance of NbSe2 for SIBs, comparing with LIBs It was investigated that NbSe2 has the high theoretical specific capacity

of 203 and 312 and activation energy of 205 and 86 for LIBs and NIBs, [41] respectively Two-dimensional MoS2 polytypes structure was estimated as potential anode materials for SIBs with a theoretical capacity of 146 , open circuit voltages ranging of 0.75 V - 1.25 , and energy barriers of 0.28 - 0.68 eV [42] The effective behaviour of two-dimensional VS2 material was

also calculated via first-principle with a capacity of 466 , voltage of 0.48 V,

energy sodium barrier of 114 Meanwhile, VS2 bulk structure showed higher

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sodium activation energy of 300 , and voltage of 0.6 V [43] By using the density functional theory (DFT), a set of 2D transition metal dichalcogenides (TMDs) such as TiS2, ZrS2, NbS2, and NiTe2 was explored as suitable anode for SIBs with theoretical capacities of 260-339 mAh/g, open circuit voltage of 0.49-0.95 V, and low sodium ion barriers energies of 0.22-0.07 eV [44]

1.3.2.4 2D transition metal carbides/nitrides

Recently, the latest group of 2D materials 2D transition metal carbides/nitrides (MXene) is popularly studied from the successful fabrication of the MAX phase MAX phase is represented by the composition of transition metal (M), carbon and nitrogen (X), and an element (mostly from 13 and 14 group) The general formula

of MXene is Mn+1Xn (like Ti3C2, Ti3CN, Ti4C3) which are synthesized by isolate MXene from MXA precursor

MXenes in batteries: Both theoretical studies and experiments showed that

MXenes are promising candidates to be anode and cathode materials Firstly, MXenes with low formula weight has shown the high gravimetric capacities like

Ti2C, Nb2C, V2C, and Sc2C MXenes can adsorb various ion sizes, which makes them suitable for non-lithium batteries Moreover, DFT has shown the low activation energy of MXenes For example, from DFT calculation Ti3C2 is a promising lithium anode material with a theoretical capacity of 320 and ultralow activation energy of 0.07eV [45] In 2016, Mo2C was investigated with a theoretical capacity of 526 and 132 for Li and Na and Li and Na activation energy were calculated of 0.14 and 0.015 [46]

1.3.2.5 Emerging 2D materials

In recent years, some classes of inorganic 2D materials like III-V, IV-IV, and IV-VI materials have drawn much attention Among these materials, silicon

monosulfide (SiS) has been proposed by ab initio DFT as a new member of 2D

materials and has also been expected as a promising anode materials for rechargeable batteries [47]

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SiS material was first proposed as a new member in IV-VI materials Its structure was hypothesized basing on black-phosphorus and blue-phosphorus

structure in 2015 by using simulation methods [48] Silicene sulfide, a configuration

of SiS was investigated by using atomic transmutation and differential evolution

global optimization methods SiS was determined as a potential compounds for electronic devices by their good electronic properties [49, 50] Silicene sulfide

structure was studied as an anode material and showed some promising properties batteries for LIBs with theoretical capacity of 167 , OCV of 0.26 , and lithium diffusion barrier of 0.43 [49] Therefore, SiS has also expected as the potential anode material for SIBs

1.4 Purpose of thesis

Our purpose is to propose a new anode materials based on the 2D materials for SIBs For this aim, in present work, we systematically investigated the adsorption mechanism of Na on the surface of 2D SiS for various of Na concentrations and diffusion mechanism of sodium ions in SiS layer for SIBs by the DFT method Firstly, we searched the most stable configuration of NaxSiS with respect to various

Na concentrations then calculated the adsorption energies of sodium on symmetry adsorption sites The sodium diffusion mechanism was explored by finding diffusion pathways between two stable Na states along two dimensions Finally, the electronic properties of NaxSiS configurations was further investigated

high-by calculating the electron density of states (DOS) with different sodium concentration

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CHAPTER 2 METHOD

2.1 Density functional theory

Density functional theory (DFT) is the most common and successful quantum mechanical approaches to understand material properties Nowadays, it is successfully applied for simulating the binding energy of molecules in chemistry and the band structure of solid in physics DFT is a method to solving the Schrodinger’s equation of N-electron systems (N can be up to thousands) with the support of High Performance Computer (HCP) The Schrodinger equation is the basic tool to study the properties of a given material The time-independent Schrodinger equation has the operator form:

(2.1) Where H is the Hamiltonian operator, E and Ψ indicate the energy, wave function This equation can be exactly solved in the case of one nucleus and one electron

Consider a system including N nucleons of Zn charge at position * + for and M electrons at position * + for The many-body wave function become:

( ) Therefore, to calculate and simulate a quantum system, we have to solve Schrodinger equation with 3N variables, in which N is the number of electron inside the system So in the system having many electrons, the solving this equation becomes very complex and take a long calculation time, requiring High Perform Computer The Hamiltonian for the whole bulk system is:

14

Where = h/2π, h is the Planck constant, m and denote the electron mass

and the respective coordinates, and are nuclear masses and the respective coordinates, and Z is the charge of the nuclei The indices i and j number the electrons and k and l the nuclei The first term in Eq 2.2 denotes the kinetic energy

of the nuclei, the second term is the Coulomb energy between the nuclei The kinetic energy of the electrons is presented in the third term The fourth term indicates the interaction among the electrons, and the last term denotes the electrostatic interaction between the electrons and the nuclei So, Eq (2.2) can be rewritten:

(2.3) Where T, Tn are the kinetic energy of the electron and nuclei, respectively The potential energy from electron-electron repulsions, nuclear – nuclear repulsion, and electron – nuclear attraction are labeled , and , respectively Therefore, from above equations, the calculation Hamiltonian for many-body system is a complex process In practice, a series of approximation are made to reduce the complex of calculation The first of approximation methods is Born-Oppenheimer approximation, the nuclei are considered to be stationary, and Eq 2.2 has to be solved for the electrons around these stationary nuclei This allows us to remove the first term in Eq 2.2 The second term is only a constant (since the nuclear positions are known) Following the Born-Oppenheimer approximation, Eq 2.2 and 2.3 can

Through the Born-Oppenheimer approximation, the complex Hamiltonian (H)

is simplified to the electronic Hamiltonian ( ) Now, the many body

wave-15

function is depended on spin and positions of the electrons while component isfigured the positions of nuclear

2.1.1 Hohenberg - Kohn theorems

Material properties are mostly exhibited through the ground-state electronic structure, the ground-state is governed by the electrons surrounding the atomic nuclei and interactions between them Hence, if we can get hold of the real space distribution of these electrons (the electron charge density ( )) then almost all physical properties can be predicted The objective of the electronic structure calculations is to obtain the electron density This is the main idea of density

functional theory (DFT) The total number of electrons, n is:

The external potential ( ) is determined by the electronic density, besides

a trivial additive constant,

∫ ( ) ( ) , ( )- (2.8) , ( )- functional of the charge density

Where the external potential ( ) presents the interaction of the electrons Additionally, the nuclei electrons interactions of the nuclei in system are also included within the term Furthermore, the three dimensional electron density

is indicated ρ(r), , ( )-term represents the kinetic energy of the electrons and the

inter-electron interactions However, unless we know , ( )- term, we still do not have all the necessary information about the ground-state energy of a many electrons system

0

( )1 ( ) (2.10) Where denotes the orbital energy, represents the Kohn –Sham orbital, and the effective potential is symbolized Veff The effective potential is expressed as:

( ) ( ) ∫ ( )

, - ( ) (2.11) Where the ( ) represents the exchange-correlation potential, which can be related to exchange-correlation energy functional ( , ( )-) by following equation:

( ) . , (

( ) / (2.12) For an arbitrary electron charge density, there is no simple explicit expression

for the exchange-correlation energy E xc The local density approximation (LDA) and generalized gradient approximation (GGA) are simple approaches to the exchange-correlation energy

17

2.1.2 Exchange-Correlation Functionals

In Kohn-Sham equation, the exchange-correlation energy is an indeterminate component, thus an approximation is required A wide various approximation methods have been used in ordered to approach this problem Local Density Approximation (LDA), the Generalised Gradient Approximation (GGA) or Hybrids (HSE) functional approximations are recently used for the , ( )- term The LDA exchange-correlation functionals are expressed as:

, - ∫ ( ) ( ( )) (2.13)

In assumption of LDA, the exchange-correlation energy per electron,

located at point r, is equivalent to the homogeneous electron gas with density ρ(r)

Although simple, the LDA results in a good description of bond length, crystal structure, elasticity properties for various systems However, the LDA is not accurate enough for simulating chemical energies (often overestimate binding energy)

The GGA exchange-correlation functionals are expressed as:

, - ∫ ( ) ( ( )) ( )) (2.14) The GGA’s results are equivalent to LDA in atomic structure term, GGA has overcome the error of determining binding energy in LDA calculation Moreover, GGA describes bandgaps of materials with more correct compared to LDA, but it generally gets underestimation the value of band gaps

Hybrids functional are a combination of Hartree-Fock exchange energy and DFT by the following equation:

( ) (2.15) Hybrids functional is claimed to be an effective way to describe exactly band gaps of almost materials, especially semiconductor

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2.1.3 Solving Kohn-Sham Equation

The exact ground-state density of the interacting system may be achieved by solving a non-interacting problem, where the potential depends on seft-consistently on the electron density The calculation process is showed by following diagram:

Figure 2.1 An illustration of the self-consistent field (SCF)

2.2 Nudged Elastic Band method

Exploration minimum energy paths (MEPs) play a critical role for finding the mechanism of reaction, energy barriers, and phase transformations Nudged Elastic Band (NEB) method is to find MEP between known reactant and product states [53] In NEB code, a set of “image” is linearly interpolated between the known initial state and final state, and then minimizes the energy of these images The NEB calculation is performed following four steps, they are:

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 Determining initial and final states

 Drawing a linear path between them

 Creating transition states, called “images”

 Optimizing each image by analysing the forces on the geometry

To ensure the convergence of all images to the same MEP, theoretical

springs are added between images Hence, each image finds the lowest energy possible while maintaining spacing to neighboring images The force components are showed in Figure 2.2

Figure 2.2 Decomposition of force in an image [54]

Two components decompose the nudged elastic band force FNEB, including the spring force ‖ along the tangent ̂ , and the perpendicular force due to the potential

FNEB = ‖+ (2.16) ( ) ( ) ̂ ̂ (2.17)