Molecular dynamics study of the wurtzite structure

by

Kim Trong Nguyen

A thesis submitted to the Department of Chemisty in conformity with the requirements for

the degree of Master of Science

Queen's University Kingston, Ontario, Canada

October, 1997

copyright O Kim Trong Nguyen, 1997

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A bstracts

The structure of crystals with the wurtnte structure (space group P6smc) has been

studied by the method of molecular dynamics. Expenmental data on the structural

parameters is reviewed, particularly the d a ratio and the positional parameter u which

specifies the relative displacement of the cation and anion sub-lattices. Deviations of d a

and u fiom the values that give regular tetrahedral coordination around each ion have

been found experimentally to be linearly related. An earlier study of the static stability of

the wurtzite structure showed that electrostatic effects are dominant in determining the

structure. The present molecular dynamics study proceeds from an assumed potential

function with an exponential repulsion term in addition to the electrostatic potential. The

B phase of silver iodide was used as a prototype wurtzite crystal. The parameters in the

potential function were chosen so as to reproduce as closely as possible to the observed

structural parameters, using molecular dynamics calculations at constant lattice geometry.

Agreement was only found with an unrealistically small value of ionic charge.

This ihesis is dedicated

to

my parents.

Acknowledgements

1 wouid like to thank my supervisor Dr. R.J.C. Brown, for his advice and support

throughout this project. I am gratefùl for his teaching and patience.

1 would like to thank Dr. D.M. Wardlaw, for the use of his facility in the last few

months of this work. I dso thank Dr. B.K. Hunter, for his helpfûl comments on the

manuscript.

The work descnbed in this thesis could have not been accomplished without the

love. guidance. and support of my dear parents. I also extend my sincere thanks to al1 my

brothers, sisters and in-laws for their persona1 encouragement and the patient supports

they have always given me.

1 would like to thank al1 my fiends especially Xuàn for proof reading parts of the

manuscript and Manj inder for sharing his valuable experiences in wtiting dissertations.

Table of contents

I. Introduction and overview

2. ExpeRmental binary-type wurtzite crystals 2.1 Introduction 2.1 Definition of the wurtzite structure 2.2 Available experimental data for binary-type wurtzite structure 2.3 Wurtzite crystals at several temperatures and pressures 2.3.1 Nl&F wurtzite crystals at several temperatures and pressures 2.3.2 AgI wurtzite crystals at several temperatures 2.3.3 Be0 wurtzite crystals at several temperatures and pressures 1.3.4 Zn0 wurtzite crystals at several temperatures and pressures 2.4 Summary of the properties of the wurtzite crystals

3 Model of the wurtzite crystal 3.3 Introduction 3.4 Brown's model 3 2 . 1 Introduction and theory 3 2 . 2 Model and calculations 3 2 . 3 Discussion 3.3 Model used in this work

4. Molecular dynamics simulation of &Ag1 wurtzite crystal. 4.1 Introduction 4.2 Molecular dynamics simulation 4.3 The "newton" program 4.3.1. The simulation box size and cut-off radius 4.4 Interatomic potential of p-AgI wurtzite crystal 4.4.1 Potential parameters fiom the approximate model which assumes q=+1 .O. 4.4.2 Varying the potential parameters described in 4.4.1. 4.4.3 Potential parameters corn the approximate mode1 with the charges

q=M.8, H.6, H.4, M.2.

4. Conclusion and future work

References

Tables

Table Page

Unit-ceil parameter at room temperature.

Themochemical data of ZnS.

Unit-ceIl parameters of M&.F at several temperature & Pressures.

Unit-ce11 parameters of AgI at several temperatures.

Unit-ce11 parameters of Be0 at several temperatures and pressures.

Unit-ce11 parameters of Z n 0 at several temperatures.

Stationary points of the equilibrium fünction.

N W simulation for L(3000)<3000> with parameters A and a obtained fiom Equation (3- 14).

Results fiom NVT simulation for L (3000)<3000>1 using A41667 kl/mol and a=2.60k1, with the lattice constant c fixed at 7.5 10 A while lattice constant a is varied.

Results from NVT simulation for L (3000)<3000>1 using A41667 Id/rnol and a=2.60k1, with the lattice constant a fixed at 4.592 A while lattice constant c is varied.

Results fkom NVT simulation for L(3000)<3000>, a is fixed at 2.60A" while A is varied.

Results fiom NVT simulation for L(3000)<3000>1 A is fixed at 41 667 kJlmo1 and a is vaned.

NVT simulation for L(2000)<2000> with the parameters A and a obtained from equation (4- 19) using q=kû.80.

NVT simulation for L(2000)<2000> with the parameters A and a obtained fiom equation (4-20) using q=kû.60.

NVT simulation for L(2000)<2000> with the parameters A and a obtained from equation (4-2 1) using q=iF0.40.

NVT simulation for L(2000)<2000> with the parameters A and a obtained fiom equation (4-22) using q=kû.20.

Figures

Figure

2- 1

2-2

2-3

2-4

Page

A unit-ce11 of the wurtzite structure. 6

A tetrahedron in the wurtzite structure. 7

c/a vs. u for wurtzite crystals at room temperature. 1 O

TetrahedraI angles vs. u for experimentai wurtzite crystals at room 13 temperature and atmospheric pressure d a vs. u for NJ&F crystals at several temperatures and pressures. 16

c/a vs. u for Ag[ at several temperatures. 18

c/a vs. u for Be0 at several temperaures and pressures. 20

d a vs. u for Zn0 at several temperatures. 22

Contour map of A(y,u). 30

Equilibrium function for n=4, mode1 B. 3 1

Equilibnum fûnction for n=6, mode1 B. 32

Equilibriurn function for n=8, mode1 B. 33

Stable configurations of the wurtzite lattice: Comparison of equilibrium 34 function with experimental results.

Components of the stress tensor vs. the sofiness parameter a for 46 constant volume simulation as descnbed in Table 4- 1.

Components of the stress tensor vs. lattice constant a with c fixed at 48 7.5 10 A for NVT simulation as descnbed in Table 4-2. Components of the stress tensor vs. lattice constant c with a fixed at 49 4.592 A for NVT simulation as described in Table 4-3.

Components of the stress tensor vs. pre-exponential parameter A with 52 a fixed at 2.60 A-' using the results as described in Table 4-4.

Components of the stress tensor vs. softness parameter a with the pre- 53 exponential parameter A fixed at 41667 kl/mol using the results as described in Table 4-5.

Components of the stress tensor vs. softness parameter a for constant 57 volume simulation as described in Table 4-6 and Table 4-7.

Components of the stress tensor vs. softness parameter a for constant 58 volume simulation as described in Table 4-8 and Table 4-9.

Chapter 1. introduction and overview

Wurtzite is a minera1 considing of crystalline zinc sulphide. which was

discovered by the French chemist C.A. Wurtz (18 17-1884). Any crystal that has the

same arrangement of the atoms as in wurtzite is said to have the wurtnte structure. Many

crystals with the wurtzite structure have characteristic properties of piezoelectricity and

semi-conductivity. Such substances have been applied in industry for the making of

acoustic wave resonators, varistors and other electronic devices (Kamiya, 1996).

Wurtzite-type crystals are also a subject for theoretical studies (Kamiya, 1996) since the

wurtzite crystal has a simple structure. The lattice is hexagonal with lattice parameters a

and c. and there is one positional -ordinate u. Both cations and anions form hexagonal

close packed structures which are displaced £tom each other by a distance uc along the c-

axis. Each ion has a quasi-tetrahedral CO-ordination around it. A perfect tetrahedron

occurs when cla = (8/3)"* and u=3/8. At these values, the wurtzite lattice is classified as

ideal. The wurtzite structure and its parameters: a, c/a and u are defined and described in

chapter 2.

There is a linear relationship between the experimentally detennined d a ratio and

the u-parameter for binary-type wurtzite crystals (Figure 2-3): as the d a ratio increases

the u-parameter decreases. In 1967, Brown reproduced this experimental linear relation

between the c/a ratio and the u-parameter by the method of static simulation. He used an

inter-ionic potential with the repulsive term containing only nearest neighbor (anion-

cation) contributions. His calculated values seem to have the same dope as obtained by

experiment but the values are displaced negatively by a small constant amount (see

Figure 2-3 and Figure 3-5). Details of his works are descnbed in Chapter 3. At present

there is no other mode1 for the energy or bonding of the wurtzite crystal which can

predict a more accurate linear relation between the d a ratio and the u-parameter.

The binary-type wurtzite crystals behave in such a way that the bond Iengths in

the tetrahedra tend to remain the same. Deviations of clri ratio or u-parameter of the

binary-type wunzite crystals from ideal are contributed mainly from the distortion of the

tetrahedral angles. The ammonium fluoride (w) crystal behaves differently and is of

interest to us. At room temperature, the c/a ratio and the u-parameter of N&F correlate

in such a way that the tetrahedra have perfect tetrahedrai angles. Non-ideal M F

wurtzite crystal is caused mainly by unequal tetrahedral distances. (The wurtzite crystals

are discussed in detail in Chapter 2).

In this work, we embark upon the study of the wurtzite structure, pmiculariy the

c/a ratio and the u-parameter of the wurtzite structure. We wiil use a model adapted fiom

the mode1 described by Brown (Brown, 1967). f.3-Ag1 wurtzite crystal is chosen as a

prototype of the wurtzite crystal to test the proposed model. The model in this work has

an electrostatic t em and an exponential fom of repulsion with only anion-cation

contributions. The model is then tested by the method of classical molecular dynamics

simulation.

The outline for this work is as follows. First, the nature of the wurtzite crystals

£Yom the experimentally determined c/a ratio and the u-parameter is described in Chapter

2. Then a review of the model proposed by Brown together with the model used in this

work is discussed in Chapter 3. Molecular dynarnics simulation of P-Ag1 is described in

Chapter 4. The conclusion of this work is given in Chapter 5.

Chapter 2 . Crystallography of wurtzite crys tals

2.1. Introduction

Generally, a computer experiment is a computational method in which physical

processes are simulated in accordance with a given physical model. In the case of

molecular dynamics simulations, the assumed physical mode1 is represented by an

interatomic potential. This interatomic potential is usually derived from experimental

data. The more experimental data that are known, the better the physical model that can

be made, and the more reliable the output of the computer experiment. In this work. we

are seeking a better understanding of the wurtzite crystals through molecular dynamic

simulations. The first step is identifying the experimental structures and properties of the

wumite crystal; that is the focus of this chapter.

In Section 2.2, a definition of the wurtzite stmcture is given. Section 2.3

discusses the avai lable experimental data on binary-type wurtzite crystals at room

temperature and atmospheric pressure. In Section 2.4, a few examples of wurtzite

crystals at other temperatures and pressures are given.

2.2. Definition of the wurtzite structure

The wurtzite structure (Fig 2-1) has the space group P63mc (no. 186). There are

two formula units per unit cell (2=2). Cations and anions are ail in Wyckoff site b with

fractional CO-ordinates (1/3,2/3,u) and (2/3,1/3,1/2+u), where u=O for anions and u = 3/8

for cations. There is a good 3-dimensional diagram of the wurtzite structure on page 192

in (Burdett, 1995). Each ion has a quasi-tetrahedral CO-ordination (site group C3, ) around

it. The quasi-tetrahedral CO-ordination can be regarded as arising from hexagonal close

packing of anions in which half the tetrahedral holes are occupied by cations. The

tetrahedra surrounding the cations al1 point in the same direction along the c-mis, while

the tetrahedra surrounding the anions points in the opposite direction. A diagram of a

tetrahedron in the wurtzite structure is illustrated (Figure 2-2) with the following

definitions for distances and angles. Nearest axial anion-cation distance is di. Nearest

basal anion-cation distance is dz. Axial tetrahedral angle is a and basal tetrahedra! angle

is p.

The wurtzite structure can also be regarded as two interpenetrating hexagonal

close packed (hcp) lattices. One hcp lattice is made entirely of A type atoms (say anions)

and the other entirely of B type atoms (cations) and these two lattices are displaced fkom

each other by a distance uc along the c-axis; where the u parameter is the ratio of nearest-

neighbour (anion-cation) distance along the c-axis to c.

An ideal tetrahedral CO-ordination is one with 4 equidistant bond lengths and 4

equivalent angles of 109.47 about each ion. The tetrahedron c m deviate from ideality

by either by distortion of bond lengths or angles, or both. The bond lengths and bond

angles are related to the experimentally detemined parameters y=c/a and u by equations

(2- 1) to (2-4).

Using the expenmentally determined parameters a, u, and y it is possible to

calculate the distances and angles around each ion in a wurtzite lanice. From equations

(2442-4) above, it follows that for ideal tetrahedral co-ordination of each ion in an ideal

wurtzite structure ~=(8 /3)~" and u=3/8. Changes in y and u parameters of the wurtzite

structure do not cause any change in space group or site symmetry.

O anions at (113,213, 0) and (213, 113, 112)

Figure 2-1. A unit-cell of the wurtzite structure

Figure 2-2. A tetrahedron in the wurtzite structure

2.3. Experimental unit-ce11 parameters of binary-type wurtzite crystals

Many binary-type compounds are known to exist in the wurtzite crystals

(O'keeffe and Hyde, 1978). A literature search by CISTI (1 996) has found that lanice

constants a and c of twenty-six compounds have been measured, while only a few u

parameters have been experimentally detennined. The rest of the u parameters are

usually assumed to have an ideal value of 318.

Presently, according to this author's knowledge, there are only nine different

binary-type wurtzite compounds for which the u parameter has been measured. They are

AIN, ZnO, N&F, BeO, CdS, CdSe, AgI, ZnS and CuBr. Table 2-1 and the

corresponding graph (Figure 2-3) of d a as a function of u summarise available unit-cell

parameters of wumite crystals with an experimentally determined u at room temperature

and one atmospheric pressure. Since CuBr is stable only at high temperature, it is not

included in Table 2- 1 and Figure 2-3.

Figure 2-3 is a graph of d a vs. u for the experimental data in table 2-1 and also

for the equiangle structure and the equidistant structure. The equiangle and equidistant

structures can be calculated using the below equations (2-5) and (2-6). When the axial

angle a and the basal angle B of a tetrahedron in a wurtzite are equd to lO9.4f, the

wurtEte structure will be referred to as the equiangle structure. For an equiangle

structure, the d a ratio and u are related by:

When the four nearest neighbour distances are equal, the structure will be referred to as

an equidistant structure. For an equidistant structure, the d a ratio and u are related by:

AIN

Zn0

Be0

CdS

GaN

CdSe

ldeal

Ag1 ZnS ND4F

Table 2-1. Unit-cell parameters at roorn temperature.

Crystai a(A) da u a (O) P (O) Ref.

0.3821 (3) 108.1 1 1 10.80 a

a/ (Schulz & Thiemann, 1977) b/ (Kisi 8 Elcom bel 1 989) CI (Downs & Ross & Gibbs, d 985) d/ (Stevenson & Milanko & Bamea, 1984) el (Stevenson & Bamea, 1984) ff (Piltz & Bamea, 1987) g/ (Lawson et al, 1 989)

Figure 2-3. d a vs. u for wurtzite cryaals at room temperature.

Wirh the exception of ND$, Figure 2-3 shows an approximately linear

relationship between d a and u for the experimental data in Table 2- 1 : as u increases d a

decreases. The linear relationship of the experimental points lies very close to the line of

equidistant structure and away fiom the line of equiangle structure. This observation

means that according to these experimental data: d a correlates with u in such a way that

di and dz bond lengths are approximately equal to each other. Hence for binary-type

wurtzite crystals (except N&F) in Table 2-1, the occurrences of non-ideal wurtzite

stmcture is due principally to the differences in the angles a and P. Figure 2-4 is a graph of the bond angles a and P calculated fiom d a and u as a

function of u for the crystals listed in Table 2-1. Figure 2-4 shows the tetrahedral angle a

increases and the angle P decreases as u increases. The further the u parameter is away

fiom the ideal value. the more the tetrahedral angles deviate from the ideal value of

109.47".

There are other observations summarized by (O'keefe & Hyde, 1978) and others

regarding the d a ratios for the wurtzite crystals which have been made by various

authors:

1. When c/a < (8/3) and u > 3/8, the bond length (di) along the c-axis is always longer

than the bond length d2 (Mair and Bamea, 1975).

2. For most wrtzite crystals, when d a is less than (8/3)1'2 the wurtzite crystal is stable.

If d a is greater than (8/3)", the compounds will fonn stable sphalente modifications

(Schultz & Thiernann, 1977). For example, ZnS has c/a=1.6378, hence the cubic

sphalerite (B3) form i s more stable. The proof for this conclusion can be seen in the

Table 2-2 below: At the sarne temperature of 298Y ArHO and An;O for the B3 structure

is lower than for the B4 wurtzite structure.

Table 2-2. Thermochemical data of ZnS.

ZnS (T=298.15K) A&IO (kJfm01) AG0 (kJ/mol)

wurtzite (B4) -191.836 -190.137

sphalerite (B3) -205.183 -200 -403

1. Barih Thmochemicd Data of Pure Substances Pm IL pp. I69î-I69% (VCH Vaiagsgcsellschaft. 1993).

3. Decreasing d a is associated with decreasing stability of the wurtzite structure with

respect to the NaCl structure. Therefore, in a sense the wurtzite structure (B4) is

intermediate between sp halerite structure (B3) and NaCl structure (B 1 ). (O'keeffe and

Hyde, 1978).

4. Within a group of MX compounds with common M atoms, the cla ratio decreases with

decreasing ionic radius ratio: R(M)/R(X). (Fleet, 1976).

6. The d a ratio correlates with the difference of the electronegativities. The compounds

with the greatest differences show the largest departwe form the ideal cla ratio. (Jefiey

et al., 1956).

Figure 2-4. Tetrahedral angles vs. u for experimental wurtnte crystals at room

temperature and atmospheric pressure.

2.4. Wurtzite crystals at several temperatures and pressures

2.4.1. Effects of temperature and pressure upon N u wurtzite crystals

The unit-ceil parameters of N m at several temperatures and pressures are

tabulated in Table 2-3. Figure 5 is a graph of d a vs. u for M&F at several temperatures

and pressures (Adrian et al, 1969) and (Lawson et al, 1989). At a constant pressure of O

kbar, the cla ratio remains almost constant while the u parameter changes with the

temperature. At the sarne temperature of 300K, ammonium fluoride crystals at two

different pressures have a similar slope as an equidistant structure but they are displaced

by a constant arnount. It is interesting to note that at room temperature and pressure,

NH4F crystal has perfect tetrahedral angles of 109.47" for both angles a and B, but with u

parameter displaced fiom ideal by 0.0015. Hence, at room temperature and at

atrnospheric pressure, the N-F distances (dl and d l ) are not equal. This is in contrast to

Schulz's conclusion (1977) for binary compounds that "There exists a strong correlation

between the d a ratio and the u parameter for al1 wurizite-type structures; if d a decreases,

the u increases in such a way that the 4 tetrahedral distances remain nearly constant and

the tetrahedral angles are diaorted." This correlation can also be seen in Figure 2-3. The

fact that foms the wurtzite structure instead of the CsCl structure may be due to

hydrogen bonding. (Note: Al1 other ammonium halides such as W C ! , MI&, NHJ

fonn the CsCl stnicture). In fact, the IR spectmrn of N m prown (1994). Plurnb et al

(1955)l indicates the presence of hydrogen bonding. These trends rnay be correct but

more recent X-ray data on IV&F (Van Beek et al., 1996) indicate c/a=1.6 155 and u =

0.3777, which puts NI&F close to the equidistant line in figure 2-3.

Table 2-3. Unittell parameters of NHIF at several temperatures &

pressures.

Temp. Press. a (4 u cia Ref

118K Okbar 4.439(2) 0.3782(3) 1.614(1) a

77 K O kbar 4.439(2) 0.378(4) 1 -61 4(1) a

300 K 4.60 kbar 4.436(1) 0.3758(3) l.598(1) b

300 K O kbar 4.436(1) 0.3735(5) 1.614(1) b

ldeal 0.375 1 -63299

a) Adnan H. W. W.. Feil D. (1 969) Acta Cryst. A25,438.

b) Lawson A. C.. Roof R. B., Jorgensen J. O.. Morosin B., Schirber J. E., (1989) Acta Cryst. 845,212.

Figure 2-5. d a vs u for M.kF cryçtals at several temperatures and pressures.

2.4.2. Effect of temperature upon Ag1 unit-ce11 parameters

The unit-ce11 parameters of A@ at several temperatures (Yoshiasa, 1989) are

tabulated in Table 24. Figure 2 6 is a graph of d a vs u for AgI at severai temperatures.

In Figure 2-6, one cm observe that with changes in temperatures, the da ratios are

constant (within 0.001 of each other), while the u parameten spread within a range of

0.3726-0.3753. From Table 24, it is also noted that at higher temperature, the u

parameters are almost indistinguishable due to the associated errors. Except for the point

at T=297K, there is a noticeable relation: as temperature increases, u decreases and

distortions from the equidistant structure and the equiangle structure increase. In figure

2-6 and at 123Y the expenmental point lies closer to the equiangle structure than the

equidistant structure. Frorn this observation, it can be said that the bond angles (a and B) of Ag1 distort very little while distortion from the equidistant structure is larger. At

higher temperature (up to 4 13K), the distortions of the tetrahedral angles became çreater

and are within 0.4" of the ideal value of 109.47 O.

In Table 2-4, the axial and basal bond length dl and dl are within 0.0077

Angstroms of each other where the axial bond length di is greater than basal bond length

dz At higher temperature, the axial bond length di decreases while basal bond length dl

increases. As well, the basal bond length d2 becomes greater than the axial bond length

dl. According to Yoshiasa et Al (1987) this shortening of the axial bond distance with

increasing temperature is due to anharmonic vibration.

Table 2-4. Unit-cell parameters of Agl at several temperatures.

Temp a (A) u d1 (4 d2(&

123 K 4.591(1) 0.3753(1) 1.636(1) 2.819 2-81 1

297 K 4.592(1) 0.3726(8) 1.635(1) 2.803 2.816

363 K 4.591 (2) 0.3733(5) 1.635(1) 2.798 2-81 9

413 K 4.590(1) 0.3726(10) 1.635(1) 2.797 2.817 Yushiasa A.. Kato K., Kanamaru F ., Emura S..Horiuchi H, (1987). Acta Cryst. 843. 434440.

Figure 2-6. d a vs u for Ag1 at several temperatures.

2.4.3. Effect of temperature and pressure upon Be0

The unit-ceil pararneters of Be0 crystal at several temperatures and pressures are

tabulated in Table 2-5 (Hazen et al, 1986). Figure 2-7 is a graph of c/a vs u for Be0 at

several temperatures and pressures. In Figure 2-7, as the temperature increases, the d a

ratios rernain unchanged while the u parameter increases and the deviation fiom both

equidistant stmcture and equiangle structure also increases. At several different

pressures, the u pararneters seem to fluctuate while changes in the d a ratio are

insignificant. There is no correlation between pressure and the u parameter in this

experiment by Hazen (1989). The u pararneters are almost indistinguishable due to the

large error associated with each measurement as can be seen in Table 2-5. It is also

interesting to note Eom Table 2-5 that as the pressure decreases or when the temperature

increases, the nearest neighbour distances di and d2 also increase. This seems to be the

correct trend as the volume of the unit ceil should increase when the pressure decreases

or when the temperature increases.

Table 2-5. Unit-cell parameters of B e 0 at several pressures and

temperatures.

Temp Pressure a(A) u d a d t ( 4 dz(A)

R.T.

R.T.

R.T.

R.T.

R.T.

543 K

763 K

963 K

1073 K

1183 K

ldeal

1.1 GPa

2.2 GPa

3.8 GPa

4.0 GPa

5.0 GPa

O

O

O

O

O

Haten. R. M.. Finger L. W., (1986). J. Appt. Phys. 59(11), 372&3733.

Figure 2-7. d a vs u for Be0 at several temperatures and pressures.

2.4.4. Effect of temperature upon Zn0

The unit cell parameters of Z n 0 at several temperatures and pressure are tabulated

in table 2-6 (Aibertsson et al, 1989). The nearest neighbor distances di and d2 are

observed to increase with the temperature. Figure 2-8 is a graph of d a vs u for Zn0

crystal at several temperatures. At al1 temperatures, the d a ratios are very close to each

other. As for the u parameters, at 20 K and 300 K, there is no change. At higher

temperature, the u parameters increase with the temperature.

Table 2-6. Unit-cell parmeters of Zn0 at several temperatures.

Temp a(A) u d a dr(& dz(A)

600 K 3.25682(5) 0.3829(3) 1.60049(6) 1.9884 1.9745

900K 3.26480(5) 0.3841(3) 1.59869(6) 1.9811 1.9693

20 K 3.241 7(8) 0.381 9(1) 1.6003(8) 1 -9959 1 -9769

300 K 3.24992(5) 0.381 9(l) 1.60206(6) 2.0048 1.9796

ldeal 0.375 1.63299 Albertsson J.. Abrahams S. C. (1989). Ada Cryst 645,3440.

Figure 2-8. d a vs u for Z n 0 at several temperatures.

2.5. Summary

For binary-type wurtzite crystals in this section (except for N&F), the c/a ratio

shows only slight dependence in temperature and pressure and the d a ratio varies only as

much as f 0.002 fiom the value obtained at room temperature and pressure. This

represents about 0.1% change. Hence it can be concluded that for these wurtzite

stiucture: the d a ratios change very little with temperature and pressure.

The experimentally determined u parameter varied as much as f 0.003 from the

value obtained at room temperature and pressure. This represents about 1% change. In

summary, the u parameter is about 10 times more sensitive to temperature and pressure

than the d a ratios. There is an interesting observation between temperature and the u

parameter. When the u pararneter is greater than 3/8, as temperature increases, the u

parameter also increases. When the u parameter is Iess than 318, as temperature

increases, the u parameter decreases. This is observed in Figure 2-5 and 2-6.

Chapter 3. Mode1 of the wurtzite crystal

3.1. Introduction

The model of an ionic cqstal is a set of assumptions about the potential energy of

interaction between the ions. Let us consider a system containing N atoms: its potential

energy can be described by equation (3-1).

The fim term in the above equation descnbes the effect of an external field on the system

and is not considered in the present work. The second term is the pair potential and is the

most important (Allen and Tildesley, 1987). The third term describes three-body

interactions and is significant in liquids and solids. Four-body (and higher) terms are

insignificant in comparison with vz and v3 and they can be ignored.

In cornputer simulations, calculations of the sum for three-body interactions are

very time consuming. Hence the term describing three-body interactions is usually

ignored. With no extemal field afFecting the system, the potential function involves only

pair-wise potential. This pair potential is generally regarded as an effective pair potential

since it represents not only the effect of the pair wise potential but also the average effect

of the three-body potential and other many body potentials. A consequence of using this

approximation is that the effective pair potential needed to reproduce experimental data

rnay tum out to depend on the density, temperature, and etc., while the tme two-body

potential v2(rr,rj) does not (Allen and Tildesley, 1987). The pair potential is assumed to

depend only on the distance between atoms or ions.

Section 3.2 will descnbe the work done by Brown (1967) on a particular model

and stability of the wurtzite lanice, and Section 3.3 will discuss specifically the mode1

which is used for molecuiar dynamics simulation of Ag1 wurtzite crystal described in

Chapter 4.

3.2. Brown's nearest neighbour model of the wurtzite lattice

3.2.1. Introduction and theory

When a particular inter-ionic potential is assumed in a calculation, the lattice

energy has a minimum for certain values of a, y, y where a is the ce11 dimension, y is the

d a ratio and u is the position parameter. Ifa parameter in the potential is varied then the

values of a, y, u at the energy minimum, will also change accordingly. In his work

Brown (1967) studied a static model of the wurtzite lattice and calculated the changes in

the equilibrium values of y and u as a function of a parameter in the potential function.

In Brown (1967)'s calculation, the repulsive term in the inter-ionic pair potential

has the inverse power form as in equation (3-2). As the parameter n in potential (3-2) is

varied, the corresponding equilibrium values of y and u at minimum lattice energy are

calculated.

The first term accounts for the electrostatic interaction between ions where q is the charge

and r is the inter-atomic distance. The second term accounts for the repulsion between

ions. The Iattice energy per ion pair can be written as:

Let p = r/o . T hen the previous equation can be written

Let M = I p-' , which is the Madelung function,

and B = C ~ - " .

Hence the potential energy can be written as a function of cell constants as in (3-3).

(3-3) V(a , JI) = -q2a-'M&,u) + pÛnB(y, cc).

Note that the MadeIung fiinction M(y,u) and hnction B(y,u) depend only on the

"shape" of the lattice (i.e. y=da and u parameter) and do not depend on the sizi of the

lattice. This makes the process of evaluating these two functions much easier. A graph

of y=c/a vs. u for the Madelung finaion is shown in Figure 3-1.

The lanice energy is stationary when a, y and u are the solutions to the equations:

If a subscript (O) is used to show a function evaluated at a stationary point (a, yo.

UQ) defined by the equations ( 3 4 , then the following arithmetic statements are tme:

(3 -5 a) q ' ~ , = np& "-' B, .

Using (3-Sa) in (3-5b) and (3-Sc),

The values of y, and u, are determined entirely by the equations (6) , which do not

contain the parameters p and q, nor the unknown b. Hence the problem of finding a

stationary point of the lattice energy is formally reduced fiom a three variable problem

(&y, u) to a two variable problem (y, u).

A practical approach is to compute values for the function (3-7) since the

equations (3-6) also define the stationary points of F (y, u).

(3-7) F ( y , u ) = ln M(y,rc) - (1ln)ln ~ ( ~ , r c ) .

A necessary and sufficient condition for V (%y, u) to have a stationary point at (&y,,, uo )

is that F(y, u) have a stationary point at ( y , u,), and a,, be detemined by equation (3-5a)

(Brown, 1967). The function F will be referred to as the equilibriurn function.

3.2.2. Models and Calculations performed by Brown

Using the general equation (3-2) for the inter-ionic potential, calculations were

performed for the following rnodels:

Model B: The repulsive energy contains only anion-cation (nearest neighbour or NN)

contributions,

Model C: The repulsive energy contains both NN and NNN contributions. Al1 three

types of interactions occur, namely anion-anion, anion-cation and cation-cation.

These three models allow derivation of the lattice energy in equation (3) From the

FORTRAN programs which were written to calculate the functions M (y, u). B (y, u) and

F (y, u). The Ewald method (Kittel, 1956) was used in calculating the Madelung function

M (y, u). Double precision arithmetic was used throughout. The equilibrium function F

(y, u) was calculated over the range 0.3 < rr s 0.5 and 1.0 < y 5 1.8. The results are

described below:

Table 3-1 shows that the results for both rnodels B and C are very similar. This

similarity shows that the use of only nearest neighbour anion-cation repulsion is adequate

for describing repulsion interactions in a wurtzite lattice.

The graphs of equilibrium functions of Model B vs. the u parameter for the cases

n=4,6,8, are given in Figure 3-2, 3-3, and 3-4. In Figure 3-2, when n=4, there is a single

maximum at yo=1.660 and uo=0.3580. In figure 3-3, when n=6, the first maximum has YO

decreased to 1.598 and increased to 0.38 10. It can be observed that as n increases (or

softness is decreasing), the first maximum will decrease. As n increases to n=8, the

maximum disappears and becornes a shoulder to a larger peak, which locates away from

the region of ideal wurtzite Iattice.

AAer establishing these curves in a general survey, calculations were made at

close intervals in the region of the shoulder in order to identify the locus of the stable

lattice configurations; the results are listed in Table 3- 1 and plotted in Figure 3-5.

3.2.3. Discussion

The results of this work on Models B and C provided the view that the wurtzite

structure is expected to appear only in compounds containing 'sofi' ions. It was found

that if the repulsive exponent n is too large there is no stable wurtzite iattice with values

of y and u anywhere near the 'ideal' values. This instability occurs for n greater than

about 8, when the stationary point of the shoulder disappears.

The locus of the stable configurations in the y-u plane, as show in Figure 3-5,

corresponds fairly weil with the experimental data (Table 2-1 and Figure 2-3). In

particular, two qualitative features of the data are reproduced; firstly the linearity of the

locus, and secondly the restricted range of the deviations from the ' ideal' structure. The

upper limit to u of about 0.4 is provided by the disappearance of the shoulder. The

linearity of the locus is not surprising (Gehman, 1965) since the bottom of the valley in

the Madelung fùnction (Figure 3-1) is almost linear in the region of interest; the linearity

must be regarded as primarily an electrostatic effect.

Despite the apparent success of the equilibrium funaion approach, it should be

kept in mind that there is a senous inconsistency in restricting the inter-ionic potential

function to the form (2). The wurtzite lattice is polar, and this work had not taken into

account dipole-dipole and higher multipole interactions between polarizable ions; these

interactions have been shown to be of importance in calculations of the electrostatic field

gradients in wurtzite crystals (Bolton and Sho1I71964) and are likely to be of importance

in detennining the equilibrium configuration ofthe lattice.

Table 3-1. Stationary points of the equilibrium hction. (Brown, 1967)

Model B Model C

u + Figure 3-1. Contour map of A(7.u). (Brown. 1967)

u + Figure 3-2. Equilibrium function for n 4 . mode1 B. (Brown. 1967)

u +

Figure 3-3. Equilibrium function for n=6, mode1 B. (Brown. 1967)

U-+

Figure 3-1. Equiiibrium function for n=8, mode1 B. (Brown, 1967)

Figure 3-5. Stable configurations of the wurtzite lattice: Comparison of equilibrium

function with experimental resul ts. (Brown. 1 967)

3.3. Mode1 used in this work

Taking the lead fiom the work done by Brown in 1967, the model used in this

work on molecular dynamics simulation of the Agi wurtzite crystals was based on the

hypot hesis that nearest neighbour anion-cation repulsion is adequate for describing

repulsion interactions. The model also assumed funher that only anion-cation repulsion

is adequate for describing al1 repulsion interactions. The model system is a simple form

of the Boni-Mayer potential (Equation 3-8) with long-range electrostatic interaction and

short-range anion-cation repulsion, which has an exponential form. The exponential

form in Equation (3-8) was used because the molecular dynamics program "newton"

assumes this potential function. This has the disadvantage that the size and shape factors

cannot be separated as in Brown's model. (Note: Brown's model has short-range

repulsion with inverse power fom and includes only nearest-neighbor anion cation

repulsion). There are two steps in finding the potential: first an approximation, then

refinement of the potential so that it would reproduce a stable Ag1 wurtzite crystal.

The coulombic term is attractive for unlike ions and repulsive for like ions. The

second term is repulsive for al1 ions. Since only anion-cation repulsions are included,

only one value of A and one value of a are needed. The factors A and a are adjustable

parameters and are deterrnined by fitting to an experimentally observed structure.

The initial potential parameters A and a can be obtained by using the

approximated potential (3-9) obtained with only the nearest anion-cation distance. Here

is a method for approximating the initial potential parameters:

There are 4 repulsions per atom since each atom sits in a tetrahedral environment. The

Equation (3-8) can be approximated by Equation (3-9).

The Equation (3-9) gives the energy per ion pair. M is the Madelung constant (for

an ideal wurtzite crystal M=1.641), and r is the shonest inter-atomic distance between

anion and cation. A and a are potential parameters for the anion-cation interaction; a

will be referred to as the softness parameter.

Taking the first derivative of the potential(3-9) with respect to r gives Equation (3-1 0).

At equilibrium r = ro,

ro is the nearest neighbor distance, which can be obtained from crystallographic data.

Expressing the A parameter in units of Idfmol. The resuiting equation is shown below:

IV4 Me 'q' exp(rp) A = x

4(4rq)rO IOOOr,a '

For A@, q= 1 .O and ro=2.8 14 A (Burley, 1962), and so the above equation becomes:

where a has a unit of A-'. The above Equation (3- 14) has parameter A as a function of a at fixed ro=2.8 14

A. The set of data (A and a ) obtaining from this relation will be used as initial potential

parameters A and a and input into Equation (3-8) together with the charge 24.0. This

set of relations should be able to reproduce the experirnental equilibrium condition with a

nearest neighbour anion-cation distance of approximately ro=2.8 14 A.

Chapter 4 . Molecular dynamic simulations of B-Ag1 wurtzite

crystal.

4.1. Introduction

Silver iodide (A@) forms yellow crystal with M. W.=Z34.77 a.m.u. and m.p.=828

K. At atmospheric pressure, Ag1 exists in three modifications (Mellander, Lenden &

Friesel, 1981): at room temperature, two phases exist: p-Ag1 (wurtzite structure) and y-

AgI (sphalerite structure), with the latter claimed to be metastable. At temperatures

between 417K and 72% the a-phase of Ag1 exists in which iodide ions form a bcc

lattice while highly mobile silver ions are distributed over many available sites (Tubandt.

1932). Because of the highly mobile silver ions, a-Ag[ is regarded as a super-ionic

conductor. It has been studied extensively through molecular dynamics simulation

(references are listed in Catlow, 199 1).

At temperatures below 41% P-Ag1 with the wurtzite structure is stable. P-A@

has been used in photography. It has been used for cloud seeding to create artificial rain

and to prevent hail-storms (Markus & Simpson, 1964; Willoughby & al., 1985).

Introduction of P-Ag[ into clouds with super-cooled water will induce freezing. Super-

cooled water is in a metastable state and is liquid below O" C; P-Ag1 has a crystal

structure sirnilar to ice and induces freezing of supercooled water. When ice particles

fom in super-cooled clouds, they grow at the expense of liquid droplets and become

heavy enough to fa11 as rain frorn clouds that otherwise would produce none. In this

chapter, P-Ag1 will be studied by the method of molecular dynamics simulation. A

previous study on P-Ag1 using MD simulation and an inverse power repulsion potential

was done by Tallon, 1988 (Tallon, 1988).

In Section 4.2, a bief introduction of molecular dynamics simulation is given.

Section 4.3 discusses the 'newton' program used for MD simulation in this work.

Section 4.4 discusses the inter-atomic potential parameters of P-AgI and corresponding

results obtained fiom simulation.

4.2. Molecular dynamics simulation

Molecular dynamics calculations are computer simulations, which can help one to

understand a system at its atomic level and observe the dynamics of translation and

rotation of molecules in the crystal. Molecular dynamics (MD) is a t em describing the

solution of Newton's equation of motion for a set of molecules. Alder and Wainwright

did the first molecular dynarnics simulation in 1957 (Alder & Wainwright, 1957). This

section gives only brief o v e ~ e w s of molecular dynamics simulation. Detailed

discussion of molecular dynamics simulation can be found in Allen & Tildesley (1987).

Catlow & Mackrodt (1982), Frenkel & Smit (1996) and Ciccotti. Frenkel & McDonald

(1987).

In molecular dynamic simulations of crystals, a given number of particles RI) is

contained in a simulation box. The simulation box is usually taken as a block of unit-

cells where each unit ce11 is the smallest repeating unit of a crystal. The simulation box is

replicated throughout space to form an infinite lattice, hence representing a macroscopic

bulk system. This is done through the use of periodic boundary condition (Born and

Karman, 1912). Dunng the simulation, as an atom in the simulation box moves out, one

of its images will enter into the box through the opposite face. In general, the periodic

images of atoms in the original simulation box behave exactly the same as the atoms in

the original box. The number density in the central box is conserved. The program only

stores the CO-ordinates of the atoms in the central simulation box. When an atom leaves

the box by crossing a boundary, attention may be switched to the image just entering.

The motion of the pariicles is govemed by their mutual interactions which corne

corn the interatomic potential. The motion can be described by using Newton's

equations of motion. For example, consider a single particle i, the Newton's equations of

motions are:

The force for the Equation (4-2) above can be derived from the interatomic

potential. The input potential $(ri . rz , ... r~ ) is usually taken to be a sum of effective pair

potentials. as discussed in chapter 3:

This potential can be split into two parts, short-range and long-range terms:

$' is oflen expressed as an exponential repulsion with dispersion interaction:

For an ionic system, the force acting upon particle i placed at $ is:

where e, is the displacement vector ( F, - < ) and qi and qi are the charges of particles i and

Due to the slow convergence of the long-range coulombic term in real space, the

Ewald technique (Ewald, 192 1; Ashcrofi & Mermin, 1984) is ofken used to overcome this

problem. The Ewald technique transforms the summation from real space into reciprocal

space where the coulombic term can converge much more rapidly .

Knowing the force field, the simultaneous Newton's equations for a system with

N particles can be solved. This is usually done through the use of the Verlet algorithm

(Verlet, 1967). The Verlet algorithm is the most widely used method of solving the

equations of motions (4-1) and (4-2). It is based on the positions i; ( 2 ) . force6 ( 2 ) and the

position < (t - ~ t ) frorn the previous step. At is the time step, which is usuall y in the

range of femto second scale. The equation for advancing the positions in the Verlet

algorithm reads as follows:

-b

(4-7). ;(t + AI) = 2 < ( t ) - <(l - Al) + : ( t ) ( ~ l ) ~ / m, .

This equation is correct up to third order in At. The velocity of particle i can be obtained

fiom 4-8:

Initially. the particles are in an ordered lattice with given velocities of random

masnitude and direction. M e r a duration of time, the system of particles reaches

equilibrium or has the Maxwell-Boltzman distribution. Then data can be collected. With

the use of statistical mechanics (see McQuarrie, 1976), it is possible to calculate various

t hermodynamic quantities.

4.3 The 'newton" program

The program used for molecular dynamics calculations is called "newtont' and is

the sarne as the one used previously ( Lynden-Bell, Ferrario, McDonald & Salje, 1989).

The time step At used for calculating molecular dynarnics simulation is 5 fs. The

program "newton" can run under constant lattice geometry, which is usually referred to

as constant volume (NVT). It can also nin under constant pressure (NPT) simulation.

In constant volume simulation, "newton" produces results in the form of stress

tensors in addition to the total energy, average temperature and pressure, diffusion

coefficients, and the order parameters cla and u. In this work, stress tensors and the

geometnc parameters u and d a are the two main results fi-om constant volume simulation

that would be analysed carefully. For a wurtzite lattice with hexagonal syrnmetry,

o,=ow . . and a, are the only two non-zero components of stress tensors. The off-diagonal

components o , on and a,, are zero by symmetry. The component oi, of stress tensors is

defined as the jth cornponent of the force per unit area across a surface perpendicular to

the direction i. Thus a, is the x component of stress per unit area across a surface

perpendicular to the x-axis. For a more detailed explanation on stress and strain in

crystals, please see the texts by (lovett, 1994) and (Nye, 1985).

In constant pressure simulation (NPT), "newton" produces results in the fom of

the shape and size of the simulation box corresponding to the given potential. Although

the results (sizes and shapes) for a constant pressure run are much easier to visualise and

analyse than the results (stress tensors) produced by a constant volume mn, the former is

much harder to obtain since constant pressure runs in "newton" are not always stable.

When the initial configuration is not close to the equilibnum configuration, "newton"

would almost always crash. This problem does not happen when "newton" cornputes at

constant volume. It was found that NPT simulation is stable for the configuration for

which stress tensors fiom NVT simulation are less than t 0.1 kbar. Therefore, in this

work, constant volume simulation is always employed first for a particular potential. If

the xx and zz components of the stress tensor are close to zero then constant pressure

simulation was used.

The initial simulation is started from the lattice with the atomic coordinates of the

Agi wurtzite unit ce11 taken from (Burley, 1962). The initial velocities are chosen

randomly. The overall average of magnitude for the chosen velocities conforms to the

specified temperature and at the same time it must not contribute any net momentum to

the simulation box. After a penod of simulation from the starting configuration, the

structure 'relaxes' to the equilibrium structure. This 'relaxing' period is referred to as the

equilibration period and is rnonitored by analyzing the instaneous values of potential

energy. The system is considered 'equilibrate' when the instaneous potential energy

oscillate about the fifth digit of a steady mean value. The coordinates and velocities of

the last sep of the 'equilibration period' are used in the starting configuration for the next

simulation, which is referred to the 'averaging period'. The results from the 'averaging

period' are kept for analysis. In this work, the series of simulation runs are described by

using the following notations: starting fiom the lattice (L), the number of steps of

equilibration is indicated in parentheses (..), and the number of steps used for averaging is

enclosed in augular brackets <..>. For instance, given the notation L(3000)<3000>, the

first part L(3000) means equilibration of 3000 time steps from the lattice and the

averaging is done over the next 3000 time steps.

4.3.1. Box size and Cut-off radius (k) for Unewton" program.

Before doing any simulation, the simulation box size and its corresponding cut-off

radius (or L) must be known. & is the potential tmncation which sets the pair

potential to zero when q r &t. If the box size is very big then the system can probably

represent that of the macroscopic system. But having a big simulation box means very

long cornputer time. In fact, the computing time is approximately proportional to the

number of particles in the simulation box. Therefore we want to find the smallest

simulation box that can adequately represent the macroscopic system.

Consider two simulation boxes, with different sizes. For any chosen potential, we

want to know if the results produced by using the smaller box are the same as those for

the larger box. For instance, in this work, a simulation box with a dimension of 6 x 6 ~ 4

unit cells containing 288 molecules was found to be able to reproduce the same resuits as

for a box size of 8x8~6 unit cells. R, was chosen so that it equals the radius o fa sphere

that fits into the simulation box. Therefore, for the reg of the simulations of A@ wurtzite

crystal a simulation box with 6x6~4 unit cells with b t = l 1.00A will be used.

4.4 Interatornie potential of Ag1 wurtzite crystal

In this section, we will use an inter-atomic potential of the form (4-9):

and attempt to see if this potential model can reproduce the expenmental d a ratio and u

parameter for the AgI wurtzite lanice by using the rnethod of molecular dynamics

simulation. The first term in the above Equation (4-9) accounts for the long range

electrostatic interaction. The second tem describes the short range Ag4 repulsion.

Other repulsions for Ag-Ag and 1-1 interactions are assumed to be negligible.

The purpose of this section is to find out the charge q, the pre-exponential

parameter A and the sofiness parameter a of Equation (4-9) that give zero xx and u

stress fiom constant volume simulation at low temperature with AgI unit cells set equal to

their expenmental values (Burley, 1962). Al1 runs are simulated at a temperature of 5 K

unless speciQ otherwise. The reason for using low temperature is to minimize the

contribution fiom the thermal motion of the ions and maximize the contributions of the

potential(4-9) to the results.

The following method is used to find the charge q, parameters A and a at zero xx

and yy stress in an isothermal constant volume (NVï) simulation: First the initial

potential parameters are obtained from an approkimated model and are used for constant

volume simulation. If these results give non-zero xx and zz stress then these initial

parameters will be refined so that they c m produce better results.

4.4.1. Potential parameters from the approximate model which assumes q-.O.

In the first step, the approximate model (which was detailed in section 3.3)

assumes the charges q-1.0 and makes use of the experimental inter-atomic AgI distance

(Burley, 1962) to obtain the relationship between A and a as shown below:

Equation (4-10) provides a relationship between the pre-exponential parameter A and the

softness parameter a, based upon the nearest neighbor AgI interatomic distance constant

of ro = 2.814 A (Burley, 1962). A sample of several different sets of A and a obtained

from Equation (4-10) and the corresponding results obtained fiom constant volume

simulation are in Table 4-1. The components xx and zz of the stress tensor are plotted in

Figure 4-1 as functions of the sofiness parameter a. It was found that as a increases the

mean stress (20,+ 0,)/3 decreases and the difference in stress (ox. - oz) increases.

There is no point on the curve in Figure 4-lat which both the components xx and zz of

the stress tensor are less than M. 1 kbar.

Table 4-1. NVT simulation for L(3000)a000>

with the parameters A and a obtained nom Equation (3-14).

alpha

Figure 4-1. Components of the stress tensor vs. the soflness parameter a for constant volume simulation as descnbed in Table 4-1.

With one set of paran

the u-parameter were sou

A41667 idmol and a=2.60

in Table 4-2 and Table 4-3. 7

Table 4-2. Resuits fiorn N V L(3000)<3000>, using A=4 1 ( with the iattice contant c fixe!

Table 4-3. Results from NV1 L(3000)<3000>, using A 4 11 with the lattice contant a fixe1 constant c is varied.

c (A) k (kbar) oa 7.300 5.25

Figure 4-2. Components of the stress tensor vs. lattice constant a with c fixed at 7.5 10 A for NVT simulation as descnbed in Table 4-2.

Figure 4-3. Components of the stress tensor vs. the lattice constant c with lattice constant a fixed at 4.592 A for NVT simulation as described in Table 4-3.

From Figure 4-2 and Figure 4-3, it is observed that the xx and u components of

the stress tensor have approximately linear relations with the lattice constants a and c.

The xx and u cornponents of the stress tensor also have different dopes as fùnctions of

the lattice constants a and c. With this information, it was thought that it is possible to

find the lattice constants a and c of the wurtzite lattice corresponding to the potential

parameters q=1.0, A=41667 kllmol and a=2.60 A". But the lattice constants

corresponding to this potential were found to be outside the range of linearity (as in

Figure 4-2 and 4-3) and were not possible to locate. Perhaps this means that this mode1

cannot produce a stable wurtzite lattice.

4.4.2. Varying the potential parameten described in 4.4.1.

Using the point A=41667 Wmol and a = 2.60 A-' in Table 4-1, the results fiom

NVT simulation when A is varied while a is fixed at 2.60 A-' and when A is fixed at

41667 kl/mol while a is varied, are explored. The results for the former are tabulated in

Table 4-4 and Figure 4-4 for the former and in Table 4-5 and Figure 4-5 for the later. It

was found that at constant a, as pre-exponential parameter A increaseq the mean stress

(20, + 0&3 also increases but the differences in stress (o, - O,) remains the same. At

constant pre-exponential parameter A, as softness parameter a increases, the mean stress

decreases and the differences in stress remain constant. The differences between xx and

u stress rernains constant regardless of the change means that the it is not possible to find

the results at zero xx and zz stress for da=1.633 and u=3/8 in this region of A and a.

Table 4-4. Results from IWT simulation for L(3000)(3000>, a is

fixed at 2.60 A-' while A is varied. a=4.592 A and ~ 7 . 5 1 0 A.

A (kl*rnol-l) a (A-') adkbar) o,(kbar)

40000 2.60 -0.27 -5 -4

40500 2-60 0.52 -4.6

4 1000 2.60 1.3 -3.8

41500 2.60 2.1 -3.1

41667 2.60 2.4 -2.8

42000 2.60 2.9 -2.3

Table 4-5. Results from NVT simulation for L(3000)c3000>, A is

fixed at 41667 kJ*mol-' and a. is varied. a=4.592 & c=7.5 10 A.

Figure 4-4. Components of the stress tensor vs. pre-exponential parameter A with a fixed at 2.6 A-' using the results descnbed in Table 4-4.

253 258 2.63 2.68

Alpha

Figure 4-5. Components of the stress tensor vs. softness parameter a with the pre- exponential parameter A fixed at 41667 kJ/mol using the results described in Table 4- 5 .

4.4.3. Potential parameters from the approsimate model, but with the charges q e . 8 , H.6, H.4 and I0.2.

In this part, the approximate rnodel (which was derived in Section 3.3) assumes

that the charges are less than + 1.0 and makes use of the experimental inter-atomic Ag1

distance (Burley, 1962) to obtain the relation between A and a as shown below in

equation (4- 1 1) to (4- 14):

A = 364.86 x exp(2*8 w / mol, for q+.8 . (2.8 M)a

A = 205.23 x exp(2-8 'pal W / mol, for q-0.6 . (2.8 l4)a

A = 22.80 x exp(2-81~a)~ lmol , for q=f0.2. (2.8 14)-a

The resuits of constant volume simulation for the above four reIations are

tabulated in Table 4-6 to 4-9 and are plotted in Figure 4-6 and Figure 4-7. For a

particular soflness parameter a, it was observed that as the charge q decreases, the mean

stress decreases and the difference in xx and zz components of stress tensor (o, - O=)

also decrease. For a particular charge, it was observed that increasing the parameter a

would result in a decrease in the u parameter. When the charge q=+0.2, the xx and n

components of stress tensor have almost zero stress (within + 0.1 kbar) (Table 4-9 and

Figure 4-7). At a=3.50 A-' and q=+0.2, the u parameter is 0.379. This value of the u

parameter is within 1% ofthe experimental u parameter of 0.375.

Table 4-6. NVT simulation for L(2000)~2000> with the parameters A and a obtained from Equation (4- 1 1 ) using q=N.80. A (k.J*rno~') a (A-') o=(kbar) o=(kbar) u

Table 4-7. N W simulation for L(2000)~2000> with the parameters A and a obtained from equation (4-12) using q=F0.60.

_*

A (kl* mol-') u (A") odkbar) o,(kbar)

2038 1.75 2.08 1.27 0.397

Table 4-8. NVT simulation for L(2000)<2000> with the parameters A and a obtained from equation (4-1 3) using q=M.40.

A (kJ*mofl) a (A-') odkbar) o,(kbar) u

906 1.75 0.97 0.64 0.401

Table 4-9. NVT simulation for L(2000)<2000> with the parameters A and a obtained from equation (4-1 4) using q=M.20.

A (kJ*mofl) a (k') odkbar) o,(kbar) u

226 1.75 0.29 0.26 0.447

Figure 4-6. Components of stress tensor vs. the softness parameter a for constant volume simulation as descnbed in Table 4-6 and Table 4-7.

Figure 4-7. Components of the stress tensor vs. the sofiness parameter a for constant volume simulation as described in Table 4-8 and Table 4-9.

Chapter 5 . Conclusion and future work.

Chapter 2 has s h o w that the experimental c/a and u-parameter of binary-type

wurtzite crystals are linearly related. The d a ratio and the u-parameter are related in such

a way that the bond distances between the nearest neighbours (anions and cations) remain

almost constant. Non-ideal binary-type wurtzite crystals are contnbuted mainly by

unequal tetrahedral bond angles. The two parameters (da and u) for the anomalous

WI.J wurtzite crystal are related in a different way.

The linear relation between d a and u of binary-type wurtzite crystals was

reproduced by static simulation using Brown's model as descnbed in chapter 3. His

model points out three main points: first, the electrostatic interaction is dominant as

evident in the graph for the Madelung fùnction in Figure 3-1, second, the nearest

neighbour contribution is enough to describe repulsion interactions as evident in Figure

3-5, third, the repulsive potential function describing the nearest neighbour interaction

should be soft (Le. the n should be less than or equal to eight in the inverse power form of

repulsion). The nice thing about using this form of repulsion is that it cm be manipulated

mathematically as described in Chapter 3. Static simulation means simulation of atoms

in equilibrium with zero velocity. The disadvantage of this kind of simulation is that

there is no consideration of temperature or pressure.

One of the objectives of this project is to test the Brown's theory by the method of

molecular dynamics simulation. Molecular dynamics simulation allows the input of

temperature and pressure since it simulates the behavior of atoms (or molecules) in

motion, in particular atoms acted on by forces and having variable velocity. The

repulsive potential used in the MD simulation program is the exponential repulsion form.

Hence the chosen model consists of electrostatic terms in addition to the exponential

anion-cation repulsive potential. Because of the nature of molecular dynamics simulation

and the exponential repulsive potential, wurtzite crystals cannot be studied generally as in

the previous study. One system namely P-AgI was chosen as a prototype for the wurtzite

crystals and was used in molecular dynamics simulations.

Molecular dynamics simulation of P-A@ was described in Chapter 4. Obtaining a

satisfactory potential function by using molecular dynamics simulation to reproduce the

P-Ag1 proves to be a dificult task. The set of potential parameters which can reproduce

the B-AgI wurtzite ciystal are listed in Table 4-8. The charge q was found to be f 0.2.

This is unrealistic since the charge obtained fiom phonon dispersion measurement is f

0.6 (Vashishta & Rahman, 1978). The possible rasons for the low charge may be

because of the fonn of the repulsion, the additions of other anion-cation repuisions to the

nearest neighbour contributions and the exclusion of ionic polarisation. Polar-related

interactions have been s h o w to have significant contributions to electrostatic field

gradients' calculations in wurtzite crystals (Bolton & Sholl, 1964).

A fundamental quantum study of the electronic structure for P-Ag1 similar to the

study of AgCl by the density-functional pseudo-potential method (Kirchoff et. al., 1994)

is suggested. An extension of this calculation (P-AgI) to the anomalous ammonium

fluoride m) rnay then be possible.

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