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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