Potential cathode material naxvopo4 for rechargeable sodium ion batteries dft investigation

38 Figure 3.10: Activation energy profile of Na vacancy – positive polaron diffusion along the [010] direction in NaVOPO4.. 40 Figure 3.13: Activation energy profile of Na ion – negativ

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY -

LUONG HUU DUC

POTENTIAL CATHODE MATERIAL

NaxVOPO4 FOR RECHARGEABLE SODIUM ION BATTERIES: DFT INVESTIGATION

MASTER'S THESIS

Hanoi, 2018

VIETNAM NATIONAL UNIVERSITY, HANOI

VIETNAM JAPAN UNIVERSITY -

LUONG HUU DUC

POTENTIAL CATHODE MATERIAL

NaxVOPO4 FOR RECHARGEABLE SODIUM ION BATTERIES: DFT INVESTIGATION

MAJOR: NANO TECHNOLOGY

SUPERVISOR:

Dr DINH VAN AN

Hanoi, 2018

Acknowledgement

First and foremost, to my supervisor, JICA expert, Dr Dinh Van An, for his patient guidance, enthusiastic encouragement and useful critiques of this research during my two-year research

Next, I would like to express my great thank to staffs and Master Students in Laboratory of Simulation, Vietnam Japan University, for their helps and discussion

on weekly seminars when I did my thesis

Finally, I would like to acknowledge the Japanese International Co-operation Agent (JICA) and Vietnam Japan University (VJU) for their great supports

TABLE OF CONTENTS

Acknowledgement i

LIST OF FIGURES iv

LIST OF TABLES vii

APPREVIATIONS viii

INTRODUCTION 1

CHAPTER 1 LITERATURE REVIEW 3

1.1 Rechargeable batteries 3

1.2 Vanadyl phosphate family 9

1.2.1 Beta Vanadyl Phosphate (β -VOPO4) 12

1.2.2 Beta Sodium Vanadyl Phosphate (β -NaVOPO4) 14

1.3 Purposes 15

CHAPTER 2 METHODOLOGY 16

2.1 Density Functional Theory 16

2.1.1 Kohn-Sham Equation 16

2.1.2 LDA, GGA and GGA+U Methods 18

2.1.3 Hybrid Functionals Method 19

2.1.4 Pseudopotentials 20

2.1.5 Solving the Kohn-Sham Equation 21

2.2 Nudged Elastic Band (NEB) 22

2.3 Calculation Scheme 23

CHAPTER 3 RESULTS AND DISCUSSION 25

3.1 Crystal structure 25

3.1.1 Vanadyl phosphate 25

3.1.2 Sodium vanadyl phosphate 27

3.2 Electronic structure 31

3.2.1 Vanadyl Phosphate 31

3.2.2 Sodium vanadyl phosphate 32

3.3 Voltage 32

3.4 Diffusion mechanism 33

3.4.1 Positive small polaron and discharging state 34

3.4.2 Negative small polaron and charging state 39

3.5 Suggestions for further studies 42

CONCLUSION 43

Bibliography 44

List of Publications 49

Appendix 50

LIST OF FIGURES Figure 1.1: Operation of Rechargeable batteries 4 Figure 1.2: Crossing and Parallel Diffusion in LiFeMnPO4 The brown, green, blue balls indicates Mn, Fe, Li ions, respectively Arrows present the hoping of polaron 10

Figure 1.3: Total DOS and the contributions of V, P, and O to the DOS of - β VOPO4

Figure 3.2: Energy difference between five spin polarization configurations of

NaVOPO4 27

Figure 3.3: Crystal structure of NaVOPO4 The brown and green octahedrons indicate the 1NN and 2NN vanadyl groups and grey tetrahedrons present the phosphate groups Blue balls are Na ions 28

Figure 3.4: Bond distances (Å) and bond angles (o) of distorted octahedral VO6 (a) and Na6O (b) in NaVOPO4 The dash blue lines indicates the possibility of Na ion diffusion pathway 30

Figure 3.5: Density of States (DOS) of nonmagnitic VOPO4 (a, b) and ferromagnetic NaVOPO4 (c, d) obtain by GGA+U and HSE06 method The positive (negative) part indicates the up (down) spin 31

Figure 3.6: Deintercalation of one Na ion from NaVOPO4 The empty square indicates a hole formed after removal 33

Figure 3.7: The Density of States of the defect structures of NaVOPO4 obtained by (a) GGA+U and (b) HSE06 methods 36

Figure 3.8: Diffusion directions of Na ion in NaVOPO4 The red, dark green, black, violet balls indicates trace of the Na ion diffusion along the [010], [111], [100], [101] direction 37

Figure 3.9: Diffusion pathway of Na vacancy – positive polaron complex along the

[010] direction in deintercaltion The brown and green octahedra indicate the 1NN and 2NN VO6 groups to the Na vacancy, respectively The red, green and blue balls present the trace o f the crossing, single and parallel diffusion processes, respectively Curved arrows illustrate the migration directions of polaron in each EDP 38

Figure 3.10: Activation energy profile of Na vacancy – positive polaron diffusion

along the [010] direction in NaVOPO4 The relative energies of the crossing, single and parallel diffusion processes are illustrated in red, blue and green, respectively 39

Figure 3.11: Intercalation of a Na ion to VOPO4 40

Figure 3.12: The density of states of the intercalated structure of VOPO4 obtained by (a) GGA+U and (b) HSE06 methods 40

Figure 3.13: Activation energy profile of Na ion – negative polaron diffusion along

the [010] direction in VOPO4 The relative energies of the crossing, single and parallel diffusion processes are illustrated in red, blue and green, respectively 42

Figure A - 1: Investigate the cut-off energy for calculation 50 Figure A - 2: Find the K-POINT value 51 Figure A - 3: Density of States (DOS) of AFM (a, b) and FM (c, d) of NaVOPO4 by GGA+U and HSE06 The negative (positive) value of DOS indicates the down (up) spin 51

Figure A - 4: Activation energy profile of Na vacancy – positve polaron complex in

NaVOPO4 The red, blue and green curves coresspond to the crosing, parallel and single diffusion processes, respectively 52

Figure A - 5: Activation energy profile of Na ion – negative polaron complex in

VOPO4 Red, blue and green curvers illustrate the crosing, parallel and single diffusion processes, respectively 53

LIST OF TABLES

Table 1-1: Voltage of some cathode materials 6

Table 1-2: LIBs and NIBs comparison 7

Table 1-3: Activation energy (meV) of Li diffusion in Olivine material 9

Table 1-4: VOPO4 family members 11

Table 3-1: Bond lengths and bond angles of V-O and P-O in VOPO4 26

Table 3-2: Bond length and bond angle of V-O and P-O in NaVOPO4 29

Table 3-3: V-O bond lengths (Å) of the defect Na3(VOPO4)4 structure by GGA+U (HSE06) method V ions marked by 1, 2, 3 has an oxidation number of +4 (magnetic moment of 1.0μB) and V4 has an oxidation number of +5 (magnetic moment of 0μB ) 35

Table 3-4: V-O bond lengths (Å) of the Na inserted structure Na(VOPO4)4 by GGA+U (HSE06) method V ions marked by 1, 2, 3 has an oxidation number of +5 (magnetic moment of 0μB) and V4 has an oxidation number of +4 (magnetic moment of 1.0μB ) 41

Table A - 1: Investigation Hubbard type potential U values 50

APPREVIATIONS

1NN: 1st nearest neighbor

2NN: 2nd nearest neighbor

DOS: Density of States

DFT: Density Functional Theory

GGA: General Gradient Approach

LDA: Local Density Approach

LIB: Lithium ion battery

MEP: Minimum Energy Path

NIB: Sodium ion battery

NEB: Nudged Elastic Band

PP: Pseudo-potential

PAW: Projected augmented wave

USPP: Ultra-soft Pseudo-potential

INTRODUCTION

Recently, rechargeable batteries have taken up the market’s energy supplies and have been applied for portable electronic devices because of its convenience, high energy density, good performance and environment-friendly Among rechargeable batteries, the so-called Lithium ion batteries (LIBs), the typical type

of rechargeable batteries having high energy density, low self-discharge, long cycle life and fewer safety risk, have been evaluated as a great achievement and applied widely to portable electronic devices In principle, the structure of LIB consists of a cathode and an anode which connect to each other through an electrolyte In batteries, the cathode plays a crucial role to determine the overall voltage

However, in order to meet the increasing energy demand and struggle with global warming issues in circumstance that the supply of Lithium for Lithium ion batteries is limited and its cost increases steadily over years, the Sodium ion batteries (NIBs) have been recently expected to be alternative electronic storage devices In order to evaluate systematically performance of an electrode material, beside the capacity, voltage, cycle life, many researchers have pointed out that it

is needed to explore the diffusion mechanism of charge carriers in the materials, including charging and discharging processes On the cathode materials, many scientists have figured out that small polaron formation affects the diffusion of charge carriers so that in general, it raises the activation energy of diffusion Therefore, the formation of polaron and its effect on the diffusion mechanism would be an essential issue in the full exploration of performance of the cathode materials for rechargeable batteries

Vanadyl phosphate is a promising material which has been investigated for cathode of both LIBs and NIBs In experiments, several distinct structures of Sodium vanadyl phosphate NaVOPO4 have been synthesized and these materials show good voltages of around 3.5V which are suitable for cathode of Sodium ion batteries applying to portable devices, such as laptop, mobile phones Although

there are some calculations to explore the diffusion mechanism, the diffusion of Sodium in defect structure is not clear, especially effect of small polaron has not yet included In order to evaluate fully this material, we have been done calculations to simulate the optimized structure of an interesting material β-

NaxVOPO4 [x=0,1], their defect structures, their density of states (DOS), magnetic properties and voltage by using an effective calculation method, Density Functional Theory (DFT) Especially, in the first time, the formation of the small polaron and the diffusion mechanism of the Na vacancy – polaron complex in the fully charging (NaVOPO4) and discharging (VOPO4) states are carefully explored

CHAPTER 1 LITERATURE REVIEW

1.1 Rechargeable batteries

Recently, the demand of energy storage increases significantly due to the population growth and the 4th industrial revolution Based on stimulation of US Energy Information Administration (EIA), the consumption of energy all over the world would be predicted to increase from 549 Billion BTU in 2012 to 815 billion BTU in 2040 [1] In order to meet the increasingly huge demand in energy and adapt to environmental related issues, renewable energy is attractive for for further study and expand its application to real life However, the renewable energy is non-continuous resource, a demanding need of electronic devices is considered for energy storage

Rechargeable batteries are known as an electronic device which is utilized to store energy and can be used many times Due to the huge demand in energy and energy storage, the rechargeable batteries or ion batteries are attracted more because of high energy density, long cycle life, fewer safety risk and convenience for portable electronic devices such as mobile phones, laptops, notebooks, cameras, vehicle As a result, a so-called Lithium ion batteries (LIBs), a typical type of solid rechargeable batteries, dominates the electronic markets and is evaluated as a great achievement in recent decades [2] Consequently, the number of studies on rechargeable batteries increases regularly over years, and investigating novel materials for rechargeable batteries become a compatible academic field

In general, the structure of LIBs contains 3 main parts which are cathode, anode and electrolyte In principles, the operation of LIBs involves two reversible processes of charge carriers: the charging process and discharging process, as shown in Figure 1.1 In charging process, when the electrons move from the cathode to anode through the external circuit, Lithium ions diffuse inside cathode and move through the electrolyte to anode Reversibly, during the discharging

process, Lithium ions diffuse from the anode through the electrolyte back to insert

in the crystal structure of the cathode Thus, for a certain material for electrode of batteries, beside the voltage, capacity, cycling properties, safety risk, the diffusion mechanism of Lithium ion in materials would be considered in order to evaluate fully the operation of materials

In principle, in the transition metal-based cathode, the operation of ion batteries

is accompanied by the oxidation/reduction of transition metal redox couple A/B The general chemical reaction can be presented as:

The standard Gibb’s free energy of this general reaction is defined as:

G 0 = -nFE 0 = -nF(E 0 cat – E 0 anod) (1.1)

Where F is Faraday’s constant (F = 96,485 C/mol), n is number of electron transferred, E 0 is the standard electromotive force or cell potential at 25oC When the condition is not standard, the potential 𝑉𝑜𝑐 of the cell is definded by Nernst’s equation:

2

Anode:

Graphite

DC power

For the case of electrodes of semiconductors, the open circuit voltage 𝑉𝑜𝑐 (V)

of batteries is determined by equation:

𝑉𝑜𝑐 =𝐿𝑖(𝑐𝑎𝑡)−𝐿𝑖(𝑎𝑛𝑜𝑑)

Where 𝐿𝑖(𝑐𝑎𝑡); 𝐿𝑖(𝑎𝑛𝑜𝑡) are Lithium chemical potential of cathode and anode; F

is Faraday’s constant (F = 96485C/mol)

The energy density is another common parameter used to evaluate the battery system The energy density is measured by discharging at an appropriate current:

Where 𝑛 is the number of Li ion diffuse in material, M is molecular weight (g)

and F is Faraday’s constant

In the batteries, the cathode plays a core role to determine the overall voltage

of batteries The cathode materials are chosen based on the electronic properties such as redox potential of transition metal, charge cycling, capacity, safety for users, lightweight With a huge attraction from scientists and investigators, many

groups of materials have been studied for cathodes of LIBs by both experiments and calculation method such as Lithiated oxides LixMO2 (M is Mn, V)[3][4],

Li2MnO3 [5], polyanionic compounds (Olivine Phosphate LiFePO4 [6][7][8][9] Silicate LiMSiO4 (M is Fe, Mn, Co),…) [10] [11] [12], Flouro-polyanionic compounds (Flouro-phosphate LixMPO4F [13] [14] [15] [16], Flouro-sulphates (LiMSO4F where M is Fe, Co, Ni, Mn)[17][18] For different portable devices, the different rechargeable batteries are employed However, the LIBs used for rechargeable batteries generally require a good voltage in the range of 1.5 to 4 eV,

3.48 [19]

2.8 [21]

[19]

3.85 [22]

3.31 [19]

3 [23]

Phosphate

[19]

3.5 [24]

3.08 [19]

3 [25]

[19] 4.8

4.19 [19] -

[19]

4.1 [24]

3.59 [19]

4.0 [26]

A3V2(PO4)3 NASICON R3c 3.12

[19]

3.8 [27]

3.3 [28]

3.4 [27] AVOPO4

Vanadyl Phosphate Pnma

3.8 [29]

3.9 [30]

3.4 [31]

3.3 [32]

Recently, a discussion in regard to the supplying of reserve sources and high cost of LIBs has been addressed Although Li ions perform well in the diffusion process of electrode materials, it is true that its natural abundance is limited to more than 13.9 million tons as the US geological survey and Lithium element take position of 25th most abundant element over the world Then, lithium resources are predicted to be sufficient to meet needs until at least the end of this century in 2100 [33] In addition, in order to satisfy the demand of ion battery production, the cost of Lithium increase steadily over year Therefore, it is under

an obligation to find alternative materials which meet the demand of high energy density, lower cost, and abundant natural sources As reported previously, Sodium

is known as the 6th most abundant element on the Earth with a plenty in the Earth’s crust Sodium belongs group IA and period 3 in periodic table, thus, similar to Lithium ions, Sodium ions have one positive charge Although the mass and size

of sodium ion are higher than those of lithium ion (respectively, rNa=0.98Å;

mNa=23g/mol and rLi=0.69Å, mLi=6.9g/mol), sodium ions would be still small enough so that it can easily diffuse in material Moreover, Sodium materials have lower cost than Lithium materials (price of LiCO3 fluctuated from 4.11 to 4.19

€/kg-1 which is about 10 to 50 times than that of NaCO3 of 0.07-0,37 €/kg-1)[34] The price of Li2CO3 in 2017 increases continuously approximately 61% than in

2016 [19] Therefore, Sodium materials are called upon to replace Lithium materials and turn point in order to struggle with problems As a result, NIBs have become inviting targets in electronic batteries research

Table 1-2: LIBs and NIBs comparison [34]

Together with experiments [35][36][37], a variety of attractive sodium-based poly-anion compounds have been investigated by using the computational techniques, such as Oxides NaxMO2 [38][39][40], Olivine NaMPO4 [41], Maricite NaMPO4, (where M = Fe, Mn, Co, Ni,) NASICON A3V2(PO4)2 [19] [28] pyrophosphate Na2FeP2O7[42] In general, these recent researches have shown that the entire voltage range of almost cathode material for NIBs is 2.0 – 4.5 V and slightly similar to cathodes material for LIBs with a small difference of 0.18 - 0.57

V [26] Thus, it is required to study further Sodium materials for NIBs and to improve the electrochemical performance of materials

In cathode materials based on transition metal compounds, the charging/ discharging processes lead to the oxidization/reduction of transition metal ions in the host material Many studies have pointed out the formation of small polaron and its effect on diffusion of Li/Na ions in cathode materials by both measurements and computational techniques.[43][44][45] Small polaron is quasiparticle which

is formed when a charge carrier is self-trapped by lattice distortion In 2006, Ceder

et al [7]investigated that the formation of a bound polaron attributes to the lattice distortion in LiFePO4 and both the Li ion and the small polaron would diffuse due

to a strong binding energy of 500 meV After that, Ellis et al [44] figured out that

the strong binding of the polaron and Li ion in Olivine LiFePO4 increases the significant amount of 215 meV of the overall calculated activation energy to and fully consistent with the experimental results Although the small polaron formation in cathode materials was addressed, the hoping movement of the polaron

accompanying with Li/Na ion diffusion was not described clearly In 2012, Dinh

et al [12] proposed a description of Li-ion diffusion as a process of Li -vacancy

accompanied by a small positive polaron, as indicated in Table 1-3 Three elementary diffusion processes (EDPs) of Li-ion vacancy accompanied by a bound polaron, including single, the parallel, and crossing diffusion have been explored,

as shown in Figure 1.2 The single diffusion occurs when polaron accompanied with a Li vacancy locates in a same transition metal site during the Li-ion

movement On the other hand, parallel (crossing) diffusion process takes place when polaron hops from a transition metal site to another one along (crossly) the

Li diffusion direction The entire diffusion processes of Li-ion can be presented by

a combination of elementary diffusion processes and the process requiring the lowest activation energy would be the most favorable Similarly, the polaron formation in many sodium – based materials have been continued to explore The polaron – Na vacancy complex has also described in the defect structures of the sodium – based cathode materials such as NASICON, Na3MnPO4CO3 [28] The diffusion mechanism of charge carriers in such material was also depicted by a combination of EDPs It has been found that the polaron strongly affects the activation energy of Na ion in these materials Therefore, the formation of polaron and its effect on the diffusion mechanism would be an essential issue in the full exploration of performance of the cathode materials for rechargeable batteries

Table 1-3: Activation energy (meV) of Li diffusion in Olivine material [12]

y Single process Crossing

process

Parallel process

Exp

(334)

600 (430)

643 (620)

630 [24] (y=0.55)

1.2 Vanadyl phosphate family

As expected to be an alternative material, Vanadyl phosphate (VOPO4) has been evaluated as a potential material candidate for both LIBs [47] [48] [49] and NIBs [32][50] [51] Many different structures of VOPO4, LiVOPO4 and NaVOPO4

have been found in experiments and simulation as indicated in Table 1-4 For the case of VOPO ,seven polymorphic distinct structures which contain long chain

network formed by sharing corners of distorted VO6 octahedral and connecting orthophosphate groups, [52] have been explored, including 3 types of unit cell: tetragonal (α [53], αI [54], αII, [55] δ, ω [56] VOPO4), orthorhombic (β and γ) [47] [57], and monoclinic (ε) [58] which phase transition is observed at high temperature

When Li ions are inserted into VOPO4 structure, LiVOPO4 will be formed Three distinct structures including triclinic, tetragonal (αI) [30]and orthorhombic (β) [29] were identified Previously, the relative stability was compared among these phases of LiVOPO4 by estimated the relative energy and the β phase is predicted to be the most stable with a slightly smaller energy difference of 30-60 meV/fu in comparison with the triclinic and tetragonal phases [29] Compared LiVOPO4 with LiFePO4, the redox potential V+5/V+4 is higher than Fe+3/Fe+2, then the voltage of LiVOPO4 in different phases, which is in the range of 3.55-3.95 V,

is higher than LiFePO4 (3.45 V) [59] [60]

Similar to LiVOPO4, NaVOPO4 also has 3 main structures of α, β, αI [50] [51] [32] With a theoretical capacity of 165 mAh/g for both NIBs and LIBs, the voltage of sodium vanadyl phosphate is slightly similar as that of corresponding lithium compound Recently, a research using computational method [31] explored that the activation energy of Li/Na ion in this group materials of vanadyl phosphate

is in the range of 123 and 541 meV Even though experimental and calculation data of various polymorphs of NaVOPO4 have already been reported, the role of

Figure 1.2: Crossing and Parallel Diffusion in LiFeMnPO4 The brown, green, blue balls indicates Mn, Fe, Li ions, respectively Arrows present the hoping of

the bound polaron in the diffusion mechanism of charge carriers inside any polymorphs of NaVOPO4 has still not been concerned Therefore, it is necessitated

to investigate fully the physical and chemical properties in these cathode materials and evaluate systematically their performance for sodium ion batteries

Because the β phase is considered as the most stable phase among materials

in this thesis, we chose the interesting material β-NaxVOPO4 [x=0,1] for our investigation

Table 1-4: VOPO4 family members

Compound Method a

(Å)

b (Å)

c (Å)

αI VOPO4

DFT [29] 6.13 6.13 4.20 90 90 90 157.70 - Exp

[54] 6.20 6.20 4.11 90 90 90 158.00 -

αI LiVOPO4

DFT [29] 6.25 6.25 4.52 90 90 90 176.40 3.65 Exp

[61] 6.29 6.29 4.45 90 90 90 175.90 3.70

αII VOPO4

DFT [29] 5.99 5.99 4.51 90 90 90 161.80 - Exp

[55] 6.01 6.01 4.45 90 90 90 160.90 - αII

LiVOPO4

DFT [29] 6.25 6.25 4.61 90 90 90 179.90 2.99

Compound Method a

(Å)

b (Å)

c (Å)

Exp

[32] 7.54 6.37 7.62 90 90 90 366.23 3.30

ε VOPO4

DFT [29] 7.16 6.93 7.29 90 114 90 330.50 - Exp

[58] 7.27 6.89 7.27 90 115 90 329.10 -

ε LiVOPO4

DFT [29] 7.20 7.40 7.17 90 122 90 322.30 3.77 Exp

[62] 6.73 7.19 7.92 90 91.3 117 342.00 4.06 Monoclinic

NaVOPO4

Exp [51] 6.52 8.45 7.12 90 115 90 354 3.6

δ VOPO4

DFT [29] 9.14 9.14 8.55 90 90 90 714 - Exp

[56] 9.06 9.06 8.61 90 90 90 706 -

δ LiVOPO4 DFT

[29] 9.23 9.23 8.74 90 90 90 746 3.61

1.2.1 Beta Vanadyl Phosphate (β -VOPO 4)

The crystal structure of β -VOPO4 was synthesized by mixing V2O5 with H3PO4

at high temperature (900oC) for the first time in 1972 by Calvo et al [57] This

form of VOPO4 is an iso-structure of β -VOSO4 with a formal oxidation state from +4 to +5 The X-Ray diffraction shows that the crystal is orthorhombic with a space

group of Pnma The obtained crystal parameters are a =7.770(3); b=6.143(3);

c=6.965(3) The crystal of β -VOPO4 consists of corner sharing distorted

octahedron VO6 parallel to the [100] direction with no additional bonding between these octahedrons The shortest and largest V-O bonds arrange alternately along this direction These bonds with a tilted V-O-V create zigzag chains These chains link to others by PO4 tetrahedrons which share their corners to four other V-O bonds of VO6 along the {011} plane

The electronic structure of β -VOPO4 is explored by using the Density Functional Theory [63] From the calculated Density of States (DOS), as shown in Figure 1.3, the band gap of this material is around 1.7 eV Under the Fermi level, there are 3 main blocks of band in the valence band which is mainly from oxygen states mixed with phosphorus states Above the Fermi level, from 1.5eV to 6 eV

in the conduction band, the DOS is contributed mainly by Vanadium states with a small contribution of oxygen states At higher energy level, the DOS is a Figure 1.3: Total DOS and the contributions of V, P, and O to the DOS of -

β VOPO4 [63]

combination of phosphorus and oxygen states Therefore, Vanadium states are crucial factor for the electronic properties of VOPO4

1.2.2 Beta Sodium Vanadyl Phosphate (β -NaVOPO 4)

The β -NaVOPO4 was reported for the first time using chemical sodiation at low temperature by an American scientist group [32] The orthorhombic crystal

structure with space group Pnma was confirmed by X-Ray diffraction The crystal

parameters of Sodium inserted material is slightly larger than material without sodium (a=7.539(0); b=6.374(0); c=7.621(1)), the volume of β -NaVOPO4 is slightly 10% larger than β -VOPO4 In general, the network crystal structure form

by zigzag VO6 chains and PO4 tetrahedrons of β -NaVOPO4 is similar to β -VOPO4 With larger bond lengths V-O due to the smaller oxidation number of Vanadium

V4+, the sodium is located in space between zigzag chains and phosphate groups

In electrochemical experiment, the β -VOPO4 show a discharge plateau at 3.3V [32], as indicated in Figure 1.4 The discharge curve is smoother after first 10 cycles and its cycle is up to 100 cycles Although this material shows a good theoretical capacity of about 150 mAh/g, the charge/discharge capacity is limited

to 80/60 mAh/g

The density functional theory was also employed to simulate the electronic properties and diffusion of Sodium ions in these materials [31] Although they got

Figure 1.4: Charge-discharge profiles of β-VOPO4 and chemically sodiated

β-NaVOPO4 at C/20 (8 mA/g) between 4.3 and 1.5 V [32]

the results of activation energy using GGA+U of 255 meV and 541 meV for ion migration barrier in β -NaVOPO4 and β -VOPO4, respectively, their result is questionable when the small polaron has not been included and its effect on diffusion mechanism of Na ion has not been clear

Na-1.3 Purposes

Based on the background suggested above, our purposes in this research are: (i) To explore the physical phenomenon in defect structures of these

materials such as the formation of small polaron

(ii) To Evaluate the ionic conductivity and performance of considered

materials via exploring the diffusion mechanism

To do so, we firstly investigate the crystal properties of bulk and defect structure

of β-NaxVOPO4 (x = 0, 1) and then, their electronic properties via density of state (DOS), band structures, magnetic properties and voltage by using Density Functional Theory (DFT)

In here, in the first time, evidences of small positive and negative polaron formation in distorted structures, including bond length changes of distorted octahedron VO6, magnetic moment change of Vanadium ions and new bound states appearance in the band gap of DOS, has been figured out Then the diffusion

of small polaron – Na ion complexes were simulated successfully with its effect

on the activation energy of Na-ions

time with accurate results, W Kohn and J A Pople et al proposed a new theory

to solve the Schrodinger’s equation called Density Functional Theory (DFT) which helped them to award Nobel Prize in 1998 [64] For the sake of simplify,

the word operator and the operator notations are omitted except Hamiltonian

Moreover, coordinate notation r is a combination of 3-dimensional position (x, y,

z) and spin (up and down) coordinates The one-tried electron density ρ(x, y, z),

which is defined as the number of electrons per unit volume at a given point r, is

used as the main variable to solve N electron problem Therefore, instead of calculating the 3N-dimensional equation, the number of equations reduces to n – separate 3-dimensional ones so that it is beneficial for the calculation In addition, the electron density is the observable quantity and does not depend on the number

of electrons of systems As a result, the DFT can use to treat system containing up

to thousands of electrons and made computer experiment possible for simulating materials DFT is a great achievement for both accuracy and efficiency [65]

In DFT scheme, electrons are assumed to not interact with others The electron density is expressed by a sum of set of squares of non-interacting orbitals:

𝜌(𝒓) = ∑|𝜙𝑖(𝒓)2|

𝑖

= 2 ∑|𝜙𝑖(𝒓)|2 𝑜𝑐𝑐

𝑖

(2.2)

where 𝜙𝑖 are the so-called Kohn Sam (KS) orbitals in noninteracting reference system.

If all the electron densities are considered over the whole space, then the total

number of electrons, n is:

Consider the first term, the kinetic energy can be expressed in summation form with the KS orbitals 𝜙𝑖:

The external energy, 𝐸𝑒𝑥𝑡[𝜌(𝒓)]is a functional, meaning that it is a function

of 𝜌(𝒓), which in turn is a function of r (electronic coordinates)

In non-interacting charge distribution, the Hartree potential causes by the

interaction between an electron at r and the mean electron density at r’ in the mean

field approximation, then:

𝑈𝐻(𝒓) = ∫ 𝜌(𝒓′)

Next, the Hartree energy can be expressed as:

The KS equation is derived with the corresponding KS Hamiltonian, 𝐻̂ : 𝐾𝑆

[−1

2𝛻2+ 𝑈𝑒𝑓𝑓(𝒓)] 𝜙𝑖(𝒓) = 𝜀𝑖𝜙𝑖(𝒓) 𝑜𝑟 𝐻̂ 𝜙𝐾𝑆 𝑖(𝒓) = 𝜀𝑖𝜙𝑖(𝒓) (2.13)

2.1.2 LDA, GGA and GGA+U Methods

There are several approximations for exchange correlation energy The local density approximation (LDA) is considered as a simple way which complexes system is divided into many pieces of uniform electron density with different values However, this approximation has many drawbacks including

underestimating the lattice parameters, too high adsorption energies and too low diffusion barrier energy, 50% smaller band gap, unwell description for transition metals and system containing weak hydrogen bond, and so on Then, the generalized gradient approximation (GGA), [66] which captures both electron density and its gradient at a given point, gives better results The general formula with density gradient is expressed as:

𝐸𝑥𝑐𝐺𝐺𝐴[𝜌(𝑟)] = ∫ 𝜌(𝑟)𝜀𝑥𝑐𝐺𝐺𝐴[𝜌(𝑟)]𝑑𝑟 (2.14)

If we consider the spin variables, the exchange correlation energy functional by GGA becomes:

𝐸𝑥𝑐𝐺𝐺𝐴[𝜌↓, 𝜌↑] = ∫ 𝜌(𝑟)𝜀𝑥𝑐𝐺𝐺𝐴[𝜌↑, 𝜌↓, 𝜌↑, 𝜌↓]𝑑𝑟 (2.15) GGA works well for almost system within 1~3% error It increases the lattice constants and improves the activation energy barriers

However, the GGA often fails in describing the d orbital of transition metal,

a Hubbard potential has been added to correct the calculation which consists of transition metal elements

𝐸𝑥𝑐𝐺𝐺𝐴+𝑈[𝑛(𝑟⃗)] = 𝐸𝑥𝑐𝐺𝐺𝐴+𝑈[𝑛(𝑟⃗)] + 𝐸𝑈[𝑛𝑚𝑚𝜎 ] (2.16)

is the atom-centered spin-orbital occupation of the ith atom on which the Hubbard correction is placed as spin 𝜎 and angular momentum projection m

2.1.3 Hybrid Functionals Method

The PBE0 hybrid functional is another effective method to approximate the exchange correlation by combining the exact exchange HF and a class of GGA functional, PBE:

𝐸𝑥𝑐𝑃𝐵𝐸0 = 𝑎𝐸𝑥𝐻𝐹 + (1 − 𝑎)𝐸𝑥𝑃𝐵𝐸 + 𝐸𝑐𝑃𝐵𝐸

(2.17)

By removing the compensated HF and PBE long-range exchange contribution, then we obtain:

𝐸𝑥𝑐𝜔𝑃𝐵𝐸0 = 𝑎𝐸𝑥𝐻𝐹,𝑆𝑅(𝜔) + (1 − 𝑎)𝐸𝑥𝑃𝐵𝐸,𝐿𝑅(𝜔) + 𝐸𝑥𝑃𝐵𝐸,𝑆𝑅(𝜔) + 𝐸𝑐𝑃𝐵𝐸

(2.18)

where 𝑎 is the mixing coefficient and 𝜔 is an adjustable parameter governing the extent of short-range interaction For HSE06 [67] case, 𝑎 =1/4 and 𝜔 =0.2 has been used and proved to be useful and effective to struggle with many quantum systems

∫ |𝜓𝑃𝑃(𝑟)|2𝑑𝑟 = ∫ |𝜓𝑟𝑐 𝐴𝐸(𝑟)|2𝑑𝑟

0

𝑟𝑐

To eliminate radial nodes, the peak position of the wave function is shifted

to bigger core radius rc with reduced peak height The new pseudo wave function

as flat as upside – down bowl is Ultra-soft pseudopotential (USPPs) This type of pseudo wave function has reduced amplitude, then it is expanded much easier with

a smaller number of plane waves and cutoff energy Therefore, it benefits the computation of about 10 time faster However, USPPs is not total charge densities like norm conserving pseudopotential

Projected –augmented wave (PAW) potential is a frozen core all-electron potential The wave function of PAW is separated into 3 parts:

Ψ𝑃𝐴𝑊 = Ψ𝑖𝑛𝑡𝑒𝑟+ Ψ𝑐𝑜𝑟𝑒 + Ψ𝑛𝑒𝑡 (2.20)

where Ψ𝑖𝑛𝑡𝑒𝑟 represents the valence part with the PW expansion, Ψ𝑐𝑜𝑟𝑒 is the core part projected on radial grid at center of atom Ψ𝑛𝑒𝑡 is overlapping part which is the additive augmentation By separating into 3 parts, Ψ is very close to the

all-electron wave function, then PAW give the accurate results as the all-electron full potential approach with beneficial computation

2.1.5 Solving the Kohn-Sham Equation

In order to solve the KS equations, three keywords that must be followed are self-consistency, variational principle and constraints Minimizing the energy can be achieved by a self-consistent solution The process of calculation is done

by two optimization processes: electronic minimization and ionic relaxation In order to avoid unnecessary calculation, iterative diagonalization is used

Figure 2.1: Procedure of iterative calculation

Trial ρ(𝐫) and Exc[ρ(𝐫)]

Solve Kohn – Sam Eq

Ĥ ϕKS i(𝐫) = εiϕi(𝐫) Solution: New {ϕi(𝐫)}

Calculate forces and update ion position

Converged/YConverged/N

o