MODELING AND SIMULATIONS OF DIELECTRIC MATERIALS
A Thesis
Presented to
The Graduate Faculty of The University of Akron
In Partial Fulfillment
of the Requirement for the Degree
Master of Science
Yuan Zhou
August, 2007
ii
MODELING AND SIMULATIONS OF DIELECTRIC MATERIALS
Yuan Zhou
Thesis
Approved: Accepted:
__________________________ __________________________ Advisor Department Chair Dr. Alper Buldum Dr. Robert R. Mallik
__________________________ ___________________________ Co- Advisor Dean of the College Dr. Ang Chen Dr. Ronald F. Levant
___________________________ ___________________________ Committee Member Dean of the Graduate School Dr. Jutta Luettmer-Strathmann Dr. George R. Newkome
__________________________ Date
iii
ABSTRACT
The perovskites ABO3 are the most important class of dielectric materials. The
perovskite SrTiO3 and the perovskite-related material CaCu3Ti4O12 are studied in this
work.
In the first part of this work, we find the structural temperature of SrTiO3 by
performing molecular dynamics simulations and investigate electronic and structural
properties of SrTiO3 by performing ab initio calculations. A strong chemical bonding
nature between Ti and O is found. This is responsible for the TiO6 octahedron behavior
throughout the phase transition.
In the second part, ab initio calculations on CaCu3Ti4O12 are performed. We
investigate the electronic properties of this material. An antiferromagnetic character of
CaCu3Ti4O12 is observed. Furthermore, we investigate the electronic properties of new
materials of different Ca, Cu ratios. This is the first time to perform ab initio calculations
to study Ca1+xCu3-xTi4O12. We find that with the increase of with increase of Ca in the
material, the optimized lattice constant and band gap increase, and insulator character
becomes much more pronounced in the material. This is in good agreement with the
experimental results.
iv
ACKNOWLEDGEMENTS
Many thanks to my advisors; to Dr. Alper Buldum, for offering valuable advice and
guidance in my research; to Dr. Ang Chen, for his patience, direction and help not only in
my research but also in my life. I will never forget the days working with them.
Many thanks to the professors in the Department of Physics at the University of
Akron, for their excellent lectures and generous assistance through these past few years. I
continue to benefit from their knowledge in living life.
Many thanks to my colleagues and friends in Akron, for their help and assistance.
Many thanks to everyone I met in Akron, for their warmness and hospitality which
makes me feel at home.
Special thanks to my parents for their encouragement and loving support.
v
TABLE OF CONTENTS
Page
LIST OF TABLES………………………………………………………………….…...viii LIST OF FIGURES……………………………………………………………………....ix CHAPTER І INTRODUCTION ........................................................................................................ 1
1.1 Background of SrTiO3 ........................................................................................... 1
1.2 Background of CaCu Ti O3 4 12 .................................................................................. 9
1.3 Outline of the thesis ............................................................................................ 12
ІI MODELING AND SIMULATION METHODS ...................................................... 13
2.1 Molecular dynamics ............................................................................................ 13
2.2 Ab initio (First-principles) calculations .............................................................. 17
III MOLECULAR DYNAMICS SIMULATIONS ON SrTiO3..................................... 21
3.1 Structure .............................................................................................................. 21
3.1.1 Cubic ......................................................................................................... 21
3.1.2 Tetragonal .................................................................................................. 22
3.1.3 Force field ................................................................................................. 23
3.2 Energy minimization ........................................................................................... 24
3.2.1 One unit cell and its superlattice ............................................................... 24
3.2.2 5*5*5 cell and its superlattice ................................................................... 25
vi
3.3 Dynamics simulations ................................................................................................ 26
IV FIRST-PRINCIPLES CALCULATIONS ON SrTiO3.............................................. 27
4.1 Structure of SrTiO3 .............................................................................................. 27
4.2 Cambridge Sequential Total Energy Package ..................................................... 28
4.3 Electronic structures and optical properties of SrTiO3 ........................................ 28
4.3.1 Band structure of SrTiO3............................................................................ 28
4.3.2 Total density of states of SrTiO3 ................................................................ 30
4.3.3 Charge density of SrTiO3 ........................................................................... 31
4.3.4 Optical properties of SrTiO3 ...................................................................... 32
4.4 Summary of the work on SrTiO3 ......................................................................... 33
V FIRST-PRINCIPLES CALCULATIONS ON CaCu Ti O3 4 12...................................... 35
5.1 Structure of CaCu Ti O3 4 12..................................................................................... 35
5.2 Energy minimization of CaCu Ti O3 4 12.................................................................. 36
5.3 Geometry Optimization of CaCu Ti O3 4 12 ............................................................. 38
5.4 Band structure of CaCu Ti O3 4 12 ............................................................................ 39
5.5 Density of states of CaCu Ti O3 4 12......................................................................... 40
5.6 Charge densities of CaCu Ti O3 4 12 ......................................................................... 41
VI FIRST-PRINCIPLES CALCULATIONS ON Ca Cu Ti O1+x 3-x 4 12.............................. 42
6.1 Structure .............................................................................................................. 42
6.2 Geometry optimization ....................................................................................... 44
6.3 Band structure ..................................................................................................... 46
6.4 Density of states .................................................................................................. 48
6.5 Charge density .................................................................................................... 50
vii
6.6 Summary of the work on Ca Cu Ti O 1+x 3-x 4 12 ........................................................ 51
REFERENCES ................................................................................................................ 53
viii
LIST OF TABLES
Table Page
1.1 Summary of SrTiO3 x-ray data...............................................................................3 3.1 Coordinates and charges for SrTiO3 in tetragonal phase.......................................22 3.2 Atomic data for Sr, Ti and O………………………….........................................23 5.1 The Wyckoff positions for CaCu3Ti4O12................................................................35 6.1 The Wyckoff positions for Ca2Cu2Ti4O12…………………………………..…....44 6.2 Summary of optimized lattice constant for different Ca, Cu ratio.........................46 6.3 Summary of direct band gap for different Ca, Cu ratio………………………….48
ix
LIST OF FIGURES
Figure Page
1.1 The cubic ABO3 perovskite structure .................................................................. 2
1.2 Low temperature phase of SrTiO3 ( tetragonal).................................................... 6
1.3 The MEM charge density distribution of STO at room temperature (a) (001) (b) (002) and (c) (110) plane................................................................................. 7
1.4 The MEM charge density distribution map of STO at 70 K (a) (001) (b) (002) and (c) (110) plane ................................................................................................ 8
1.5 Structure of CCTO shown as TiO6 octahedra, Cu atoms bonded to four oxygen atoms, and large Ca atoms without bonds.............................................. 10
2.1 Simulation as a bridge between microscopic and macroscopic properties......... 14
2.2 Kohn-Sham approach.......................................................................................... 18
2.3 Local density approximation............................................................................... 19
3.1 Cubic structure of SrTiO3 ................................................................................... 22
3.2 Total potential energy vs lattice constant for SrTiO3 .......................................... 24
3.3 Total potential energy vs lattice constant for 5*5*5 superlattice........................ 26
4.1 The structure of SrTiO3 built by MS Modeling .................................................. 27
4.2 The calculated energy band structure of SrTiO3 ................................................. 29
4.3 Total density of states of SrTiO3 ......................................................................... 30
x
4.4 Charge densities of SrTiO3 (a) Sr-O plane (b) Ti-O plane .................................. 31
4.5 Refractive index and extinction coefficient of SrTiO3........................................ 33
5.1 The structure of CaCu3Ti4O12 built by MS Modeling......................................... 36
5.2 Energy minimization of CaCu3Ti4O12................................................................. 37
5.3 Bond lengths and angles of CaCu3Ti4O12 after Geometry Optimization............ 38
5.4 Band structure of CaCu3Ti4O12 ........................................................................... 39
5.5 Density of states of CaCu3Ti4O12........................................................................ 40
5.6 Spin-up and spin-down change densities of CaCu3Ti4O12.................................. 41
6.1 The structure of CuTiO3...................................................................................... 43
6.2 The structure of Ca2Cu2Ti4O12 ............................................................................ 43
6.3 The structure of Ca3CuTi4O12 ............................................................................. 43
6.4 The structure of CaTiO3 ...................................................................................... 43
6.5 Band structure of CuTiO3 ................................................................................... 47
6.6 Band structure of Ca2Cu2Ti4O12.......................................................................... 47
6.7 Band structure of Ca3CuTi4O12 ........................................................................... 47
6.8 Band structure of CaTiO3.................................................................................... 47
6.9 Density of states of CuTiO3 ................................................................................ 49
6.10 Density of states of Ca2Cu2Ti4O12 .................................................................... 49
6.11 Density of states of Ca3CuTi4O12 ...................................................................... 50
6.12 Density of states of CaTiO3 .............................................................................. 50
6.13 Charge densities of Ca2Cu2Ti4O12 (a) Ca-Cu-O (b) Ti-O ................................. 51
1
CHAPTER І
INTRODUCTION
Many experimental and theoretical methods are performed to study the structural
phase transitions of the perovskites ABO3. SrTiO3 is an important member of the
perovskites. The purpose of this work is to find the structural temperature of SrTiO3 by
molecular dynamics simulations and to examine the ground state properties of SrTiO3
using ab initio calculations in order to understand the chemical bonding nature of SrTiO3,
which is very important to explain the TiO6 octahedron behavior throughout the phase
transition.
Another perovskite-related material, CaCu3Ti4O12 has drawn much interest due to its
unusual dielectric behavior. Another part of this work is to examine the ground state
properties of this material using ab initio calculations. We also investigate the electronic
structures of Ca1+xCu3-xTi4O12 (x=-1, 1, 2, 3). This is the first time to study these
materials using ab initio calculations.
1.1 Background of SrTiO3
The perovskites ABO3 are the most important class of ferroelectric materials. At
high temperatures, they all share the simple cubic structure, with monovalent or divalent
cation A at the cube corners, penta- or tetravalent metal B at the body centers, and O
atoms at the cube face centers (see Figure 1.1). This structure can also be thought of as a
set of oxygen octahedra arranged in a simple cubic pattern and connected together by
shared O atoms, with the A atoms occupying the spaces in between. The latter picture of
corner-shared octahedra has great structural utility, because most structural
transformations observed in perovskites can be characterized as collective rotations of the
octahedra.
Figure 1.1 The cubic ABO3 perovskite
The most interesting aspect of the cubic perovskite structure is that as the
temperature is reduced, it can display a variety of structural phase transitions, ranging
from ferroelectric and antiferroelectric to non-polar antiferrodistortive in nature. [1]
SrTiO3 (STO) is an important member of the perovskites, which possesses
non-ferroelectric structural phase transitions at about 105 K. [1] Different experimental
methods involving electron paramagnetic resonance [2], ultrasonic measurements [3, 4],
nuclear magnetic resonance [5], and the polarizing microscope [6, 7] were performed to
show that there was a phase transition around 110 K. Lytle [8] performed x-ray
diffractometry on single-crystal STO over a large temperature range, from 4.2 K to 300 K.
2
The lattice parameters were determined as a function of temperature. The data was
summarized to show that as the temperature goes down, STO changes to lower symmetry
structures (as shown in Table 1.1).
Table 1.1 Summary of SrTiO3 x-ray data [8]
Besides a great deal of experimental research, a wide range of theoretical studies
were performed to find and explain the structural phase transitions in STO. Pytte and
Feder [9] studied phase transitions in perovskites with the help of a model Hamiltonian.
They found that three degrees of freedom (15 degrees of freedom per unit cell in the
perovskite structure) are directly connected with the structural phase transitions. For STO,
the phase transition from the cubic to the tetragonal phase involves the rotation of the
octahedron about a cube axis. The predictions of their Hamiltonian model qualitatively
agree well with the experimental results except for the magnitude of the angular
distortion. This disagreement suggests that additional anharmonic interactions are needed
to be taken into account. One year later, Feder and Pytte [10] included the interaction
between soft-mode coordinates and the strains into their Hamiltonian model. Good
3
4
agreement was obtained with all the experimental results this time.
Zhong and Vanderbilt [11] claimed that the previous phenomenological model
Hamiltonian approach [9, 10] had been limited by oversimplification and ambiguities in
interpretation of experiment. They performed first-principle density-functional
calculations (a more accurate method) to show that the coexistence of both
antiferrodistortive (AFD) and ferroelectric (FE) instabilities is very common in cubic
perovskite compounds. Monte Carlo (MC) simulations were performed on the STO cubic
lattice with periodic boundary conditions. The system was found to adopt the cubic
structure at high temperature. As the temperature is reduced, a transition to an AFD
tetragonal structure occurs at 130 K, a second transition occurs at 70 K to a FE tetragonal
structure, and the system transforms to the low-symmetry monoclinic structure at 10 K.
They also found that increased pressure enhances the AFD instability while suppressing
the FE one.
As we know, a first-principles approach can study the zero-temperature properties of
the materials. However, it is important to see whether one can understand such features as
the phase transition sequence and transition temperatures on a material-specific basis.
Zhong and Vanderbilt [12] extended the ab initio effective Hamiltonian treatment of first
principles to include quantum fluctuations. Path-integral MC quantum simulations were
applied to STO to find that the AFD phase transition temperature decreases from 130 K
[11] to 110 K, in excellent agreement with the experimental result of 105 K. They also
found that the quantum fluctuations completely suppress the FE phase transition, i.e. the
FE phases disappear. This implies that quantum effects on the FE instability are much
stronger than on the AFD instability in STO. Since the structural differences and energy
barriers between the cubic structure and the distorted structures are very small, a rough
estimate of the importance of quantum fluctuations can be obtained from the
Heisenberg uncertainty principle / 2p qΔ ⋅Δ ≥ h , or equivalently,
2 /(8 )E mΔ ≥ Δh 2q (1.1)
Where,
qΔ is the structural difference between phases, and EΔ is the energy uncertainty,
which may prevent the occurrence of the distorted phase if it is larger than the classical
free-energy reduction. It is known that the AFD instability involves only the motion of
oxygen atoms, while the FE instability involves mainly Ti atoms. The structural change
involved in the FE distortion (0.1 a.u. for Ti in SrTiO3) is much smaller than for the AFD
distortion (0.3 a.u. for O). Therefore, mΔq2 turns out to be three times larger for the AFD
case, even though Ti atoms are three times heavier than the oxygen atoms. According to
Equation (1.1), the effect of the quantum fluctuations will be more significant for the FE
case.
The structural phase transition of STO at ~110 K is represented by the rotation of the
TiO6 octahedron. [13] The tetragonal structure of STO at low temperature is shown in
Figure 1.2. The oxygen atoms are all equivalent in the cubic phase, while they can be
distinguished as two kinds O(1) and O(2) in the tetragonal phase. It should be noted that
O(2) atoms are no longer located at the face centers. The mechanism of the phase
transition is understood by both neutron [15] and X-ray [16] diffraction, however why the
rotational mode of the TiO6 octahedron is unstable at 110 K is still unclear.
5
6
Figure 1.2 Low temperature phase of SrTiO3 (tetragonal) [14]
Therefore Ikeda et al. [14] applied the Maximum Entropy Method (MEM) to X-ray
diffraction data to understand the chemical bonding nature of STO in both phases and to
reveal the structural changes at the electron level which occur throughout the phase
transition. MEM charge density maps of STO at room temperature are shown in Figure
1.3.
7
From Figure 1.3, both Sr and O atoms are found to be isotropic and no covalence
electrons are found between them. A rather strong covalent bond is found to exist
between the Ti and O atoms. MEM charge density maps of STO at 70 K are shown in
Figure 1.4 to compare with Figure 1.3.
Figure 1.3 The MEM charge density distribution of STO at room temperature (a) (001) (b) (002) and (c) (110) plane.
Contour lines are drawn from 0.4 at 0.2 eÅ3 intervals. Four unit cells are shown. [14]
Figure 1.4 The MEM charge density distribution map of STO at 70 K. (a) (001) (b) (002) and (c) (110) plane. Contour lines are the same as Figure 1.3.
Two tetragonal cells are shown. [14]
From Figure 1.4 (a), both the Sr and O(1) atoms are fairly isotropic and no
covalence electrons are found between Sr and O(1). From Figure 1.4 (b), there is a strong
covalent bond between Ti and O(2). The charge density distribution around the oxygen
atoms is not symmetrical with respect to the Ti-Ti lines. There is a little positional shift of
8
9
the O(2) atom in the direction of the arrow. In Figure 1.4 (c), the covalent bond between
Ti and O(1) is observed to be similar to that in Figure 1.3 (c). Therefore, it shows that
there is no significant change of chemical bonding through the phase transition. The
phase transition of STO at 110 K is purely due to the TiO6 octahedron, which seems to be
maintained by the strong Ti-O covalent bond.
1.2 Background of CaCu3Ti4O12
CaCu3Ti4O12 (CCTO) has drawn much recent interest due to its unusual dielectric
properties. Subramanian et al. [17] first observed the unusually high dielectric constant of
CCTO. This compound possesses a large dielectric constant about 12,000 at 1 kHz, which
is nearly constant over the temperature range 100-380 K. [18] It is also found that the
dielectric constant of this material is frequency dependent, decreasing with increasing
frequency. One curious result is that the dielectric constant drops as low as 100 when
cooling below 100 K.
The structure of CCTO was first determined from neutron powder diffraction
data. [19] The crystal structure is shown in Figure 1.5. The structure of CCTO can be
viewed as a perovskite-related structure, in which four ATiO3 units comprise the primitive
cell (where A is either Ca or Cu). Because there are 5 atoms in an ATiO3 unit cell, the
structure in Figure 1.5 contains 40 atoms. This is a doubled conventional cell, which is
also the primitive cell of the spin structure. The CuO4 plaquettes are nearly square,
because the bonds between Cu and O are all the same length, while the angles between
bonds differ slightly from 90 degrees. In a similar way, the distances between Ti and its
six neighbors are the same, but the angles deviate slightly from 90 degrees. Every oxygen
atom belongs to a single planar CuO4 plaquette and to the two connected tilted TiO6
octahedra.
Figure 1.5 Structure of CCTO shown as TiO6 octahedra, Cu atoms bonded to four oxygen atoms, and large Ca atoms without bonds. [17]
A material with giant dielectric constant is always classified as a relaxor or a
ferroelectric, but empirical evidence tends to exclude CCTO from either category.
Ramirez et al. [18]
excluded ferroelectricity as a cause of the unusual high dielectric
response by X-ray diffraction and thermodynamic data. Instead they explained their
results by a relaxor model. However, nanodomains or disorder effects, which are
common to relaxor materials, are not observed. Neither superstructure peaks nor strong
diffuse scattering are present in diffraction experiments. [20]
Many researchers have presented theoretical insights into intrinsic lattice of CCTO.
10
11
He et al. [20] performed first-principles calculations within the local spin-density
approximation (LSDA) on CCTO to study the structural and electronic properties. Their
calculations appear to limit certain intrinsic mechanisms to explain the enormous,
low-frequency dielectric response. Furthermore, they suggested that increased attention
should be given to extrinsic effects. An extrinsic mechanism which is associated with
defects, domain boundaries, or other crystalline deficiencies, was initially proposed by
Subramanian et al. [17] He et al. also found CCTO to be stable in a centrosymmetric
crystal structure with space group Im3, arguing against the possibility that CCTO is a
conventional ferroelectric or relaxor.
He et al. [21] extended their previous work on CCTO by carrying out a parallel study
on the closely related material CdCu3Ti4O12 (CdCTO). Replacing Ca with Cd is found to
leave many calculated quantities largely unaltered. This indicates that both CCTO and
CdCTO possess similar intrinsic structural, vibrational and dielectric properties. As for
the lattice contributions to the static dielectric constant, the discrepancy between their
computations and those measured at frequencies below the Debye cutoff range, reinforces
the conclusion that some extrinsic mechanism is likely to be the source of the large
dielectric constant present in both materials.
Johannes et al. [22] performed first-principles calculations with a full-potential
linearized augmented plane-wave method (FLAPW) within the local spin-density
approximation (LSDA). Their results reproduce the observed antiferromagnetic (AFM)
insulating character of CCTO, and show a similar density of states as that presented by
He et al. [20] They applied the superexchange picture of magnetism to CCTO, and
suggested that extremely long-range interactions are important and perhaps even
12
responsible for the AFM order.
Li et al. [23] applied FLAPW method within the generalized gradient approximation
(GGA) to investigate the basic electronic and magnetic properties of CCTO. Their
calculated indirect band gap of 0.51 eV and direct band gap of 0.58 eV within GGA are
much closer to the experimental optical gap (≥ 1.5 eV) [24] than those in He et al.’s work
[20], but still underestimate the experimental data. A metastable state is found to be
ferromagnetic (FM) and semiconducting, indicating semiconducting behavior and an
AFM-FM transition in CCTO. Besides the Ti cation expected to be involved in the
magnetic coupling [22], a possible oxygen path is proposed in the superexchange
interaction between Cu spins.
1.3 Outline of the thesis
The background knowledge of molecular dynamics simulations and ab initio
calculations is discussed in Chapter II. The molecular dynamics simulations and ab initio
calculations on SrTiO3 are discussed in Chapter III and IV, respectively. The electronic
properties are provided. The strong covalent bonding between Ti and O is found to
explain the TiO6 octahedron behavior throughout the structural phase transition. In
Chapter V and VI, we present the results of ab initio calculations on Ca1+xCu3-xTi4O12, i.e.
CuTiO3, CaCu3Ti4O12, Ca2Cu2Ti4O12, Ca3CuTi4O12 and CaTiO3. With the increase of Ca
in the material, the optimized lattice constant and band gap increase, and insulator
character becomes much more pronounced.
13
CHAPTER ІI
MODELING AND SIMULATION METHODS
2.1 Molecular dynamics
In general, molecular systems, which consist of a great number of particles, are so
large that it is impossible to find their properties analytically. Molecular dynamics (MD)
simulation serves as a good method to solve the problem numerically. Nowadays MD
simulations, first introduced by Alder and Wainwright in 1950, are a widely used method
to study time-dependent properties of a molecular system in various material-related
fields. [25] In physics, MD is used to observe the dynamics of atomic-level phenomena,
such as thin film growth and ion-subplantation. In biology, the MD method can provide
much detailed information on the fluctuations and conformation changes of proteins and
nucleic acids. In chemistry, MD simulation is a useful approach to help in the efficient
synthesis of compounds and in drug design.
In a MD simulation, the macroscopic properties of a system, such as pressure,
energy and heat capacities, are deduced from microscopic information. [26] Therefore MD
simulations act as a bridge between the macroscopic world and microscopic length and
time scales (shown in Figure 2.1). The connection between macroscopic properties and
microscopic information is made by mathematical expressions which relate the
macroscopic properties to the distribution and motion of the atoms. MD solves the
dynamics of a classical interacting many-particle system by solving the equations of
motion of the particles.
Figure 2.1 Simulation as a bridge between microscopic
and macroscopic properties [26]
Newton’s equation ( i iF m a=r
ir ) is used to simulate atomic motion. For each atom i in
a system of N atoms, is the total force acting on it by all the other particles, miFr
i is the
atom mass, and is its acceleration. This is the so-called N-body problem. Since the
acceleration can be expressed as
iar
ar2
2i
id radt
=r
r , Newton’s equation is expressed in the
following form,
2
2i
i id rF mdt
=rr
(2.1)
The force can be considered as the derivative of the potential energy with
respect to the change in the atom’s position.
Fr
i iF U= −∇r r
(2.2)
14
We need to find the potential energy U to calculate the force from Equation 2.2.
If is known, with the help of Equation 2.1, we can compute trajectories. (The
coordinates and velocities of all the particles are called the trajectory of the system.)
iFr
iFr
In MD simulations, a force field is used to describe the time evolution of bond
lengths, bond angles and torsions, also the non-bonding van der Waals and electrostatic
interactions between atoms. [27] A simple force field functional is shown in Equation 2.3.
2 2
12 6
1 1 0
( ) ( ) ( )2 2
(1 cos( ))2
4 [( ) ( ) ]4
i ii o i o
bonds angles
n
torsions
N Nij ij i j
iji j i ij ij ij
k kU r l l
V n
q qr r
θ θ
ω γ
σ σε
πε= = +
= − + −
+ + −
⎛ ⎞+ − +⎜ ⎟⎜ ⎟
⎝ ⎠
∑ ∑
∑
∑∑
r
r
(2.3) [27]
( )U rr denotes the potential energy which is a function of the positions of
particles. The first two terms in Equation 2.3 model the interactions between pairs of
bonded atoms, modeled by a harmonic potential. The third term is a torsional interaction
potential. The last term is the non-bonded term. The non-bonded term is modeled using a
Coulomb potential term for electrostatic interactions and a Lennard-Jones potential for
van der Waals interactions.
N
[27]
From a potential energy forcefield, MD can solve the classical equations of motion
for a system of N interacting atoms. Combining Equation 2.1 and 2.2, we can rewrite
Newton’s equation,
2
2i
i id rU mdt
−∇ =rr
(2.4)
Given the initial coordinates and velocities at time , the positions and velocities at t
15
time + Δ t are calculated. The initial coordinates are determined in the input file or
from a previous operation such as energy minimization, while the initial velocities are
randomly generated at the beginning of each run based on the desired temperature.
Therefore, dynamics runs can not be repeated exactly except if the same speed is used for
the random number generator.
t
Numerous numerical algorithms have been developed for integrating the equations
of motion (Equation 2.4), such as the the Verlet algorithm, the Leapfrog algorithm, the
Verlet velocity algorithm and Beeman’s algorithm. Verlet algorithm is used to our MD
simulations. All the integration algorithms assume the positions, velocities and
accelerations can be approximated by a Taylor series expansion, [25]
2
2
1( ) ( ) ( ) ( ) ,21( ) ( ) ( ) ( ) ,2
( ) ( ) ( ) ,
r t t r t v t t a t t
v t t v t a t t b t t
a t t a t b t t
δ δ δ
δ δ δ
δ δ
+ = + + +
+ = + + +
+ = + +
K
K
K
(2.5)
where and represent the position, velocity and acceleration respectively. ,r av
When choosing which algorithm to use, we should consider whether the algorithm
satisfies the conservation of energy and momentum, whether it is computationally
efficient and whether it permits the use of a relatively long time step for integration.
The MD simulation acts not only as a bridge between macroscopic and microscopic
as I mentioned above, but also as a bridge between theory and experiment. Sometimes we
can conduct a simulation and compare the theoretical results with experimental results to
test a theory or a model. A better advantage of MD simulation is that we can carry out
simulations on the computer that are difficult or impossible to conduct in the laboratory.
16
17
2.2 Ab initio (First-principles) calculations
As we know, a material is composed of atoms bound by chemical bonds, which are
simply interactions between electrons. These interactions can be described by the laws of
quantum physics. This means that all material properties (chemical, mechanical, electrical,
magnetic, optical, thermal…) can, in principle, be predicted from the atomic number and
mass of the atomic species involved, with the aid of quantum physics. This is precisely
the basic idea of first principles calculations.
An ab initio calculation is performed from first principle. The Latin term ab initio
means “from the beginning”. It implies that the calculation starts directly at the level of
established laws of physics and does not depend on empirically derived parameters.
Density functional theory (DFT), for which Prof. Walter Kohn was awarded the
1998 Nobel prize in chemistry, is considered to be an ab initio method to determine the
molecular electronic structure. The fundamental idea of DFT is that any property of a
system of many interacting particles can be viewed as a functional of the ground state
charge density. DFT, which maps the original many-electron problem into an equivalent
single-electron problem, is an extremely successful approach for the description of
ground state properties of metals, semiconductors, and insulators.
The Kohn-Sham method, the most common implementation of DFT, has provided a
way to make useful approximate ground state functions for real systems of electrons. The
Kohn-Sham ansatz replaces the intractable many-body problem of interacting electrons in
a static external potential to a tractable problem of non-interacting electrons moving in an
effective potential. (shown in Figure 2.2)
Figure 2.2 Kohn-Sham approach
The Kohn-Sham method assumes that the ground state density of the original
interacting system is equal to that of some chosen non-interacting system.
( ) ( )ksn r n r= (2.6)
The ground state energy functional can be expressed in the form
2
1
( ) ( ) ( ) ( ) ( )
1( )2
( ) ( ')( ) ''
( ) ( ) ( )
KS KS Hartree ext xc
n
KS i ii
Hartree
ext ext
E n T n E n E n E nwhere
T n
n nE n d d
E n V n d
ψ ψ=
= + + +
= − ∇
=−
=
∑
∫
∫
r r r rr r
r r r
(2.7)
( )xcE n is the so-called exchange-correlation energy.
The Kohn-Sham variational equation, shown in Equation 2.8, is similar in form to
the time-independent Schrödinger’s equation, except that the potential experienced by the
electrons is formally expressed as a functional of the electron density.
21[ ( ; ) ( ; ) ( ; )] ( ) ( )2 Hartree GS ext GS xc GS i i iV n V n V n ψ εψ− ∇ + + + =r r r r r (2.8)
In practice, the exact functional of the electron density is unknown. Practical 18
applications of DFT are based on approximations for the so-called exchange-correlation
potential, which describes the effects of the Pauli principle and the Coulomb potential
beyond a pure electrostatic interaction of the electrons.
The most widely used and simplest approximation of the exchange-correlation
energy functional is the local density approximation (LDA) which assumes that the
density can be treated locally as an non-interacting homogeneous electron gas (shown in
Figure 2.3), and the exchange-correlation energy at each point in the system is the same
as that of an uniform electron gas of the same density.
19
Figure 2.3 Local density approximation
inhomogeneous system at point r with local density n(r)
homogeneous electron gas with same density n(r)
The exchange-correlation energy can be expressed in the form
( ( )) ( ) ( ( ))xc xcE n n nε= d∫r r r r (2.9)
where ( ( ))xc nε r is an energy per electron at point r.
The local, energy-independent exchange-correlation potential V n is the
functional derivative of ,
( ( ))xc r
( ( ))xcE n r
( ( ))( ( ))( )
xcxc
E nV nn
∂=
∂rr
r (2.10)
The LDA functional within the Kohn-Sham approach has been proved to be an
accurate, practical way to investigate the solid-state properties of materials.
There are two methods to deal with the external potential , which is complicated
in real circumstances. One is called the all-electron calculation method, the other is the
pseudopotential method. The primary application of the pseudopotential is to replace the
extV
20
strong Coulomb potential of the atomic nucleus and the effects of the tightly bound core
electrons by an effective ionic potential acting on the valence electrons. Most
pseudopotential calculations are based on norm-conserving potentials, in which
pseudo-wavefunctions are normalized and the total pseudocharge inside the core matches
that of the all-electron wave function. This method provides a way to construct
potentials that are successfully applicable to calculations on molecules and solids.
However, for many cases, such as O 2p or Ni 3d orbitals, a very large plane-wave
basis-set size is needed to satisfy the norm-conserving condition. This leads to more
calculations than those in other cases. Therefore, Vanderbilt [28] developed ultrasoft
pseudopotentials which create much smoother pseudofunctions and use considerably
fewer plane-waves for calculations of the same accuracy. This method is always used in
the study of ferroelectrics.
21
CHAPTER III
MOLECULAR DYNAMICS SIMULATIONS ON SrTiO3
Molecular dynamics simulations were carried out using Cerius2 Version 4.10.
Cerius2 provides a wealth of tools for applications in life and materials science modeling
and simulation. It offers capabilities for modeling materials structure, properties, and
processes, with applications in catalysis, crystallization, and polymer science.
3.1 Structure
Both cubic and tetragonal structures were built. The cubic structure was used to get
energy minimization and the tetragonal structure was used in dynamics simulations.
3.1.1 Cubic
The space group of STO is pm3m. The Wyckoff positions are Sr (0, 0, 0), Ti (0.5,
0.5, 0.5) and O (0.5, 0.5, 0). The charges are fixed to be +2, +2.2, -1.4 for Sr, Ti and O,
respectively, due to the bond covalency— the Ti-O bond in real crystals has both ionic
and covalent character. [29] The crystal structure is shown in Figure 3.1.
The lattice constant is chosen to be a=3.905 Å
Figure 3.1 Cubic structure of SrTiO3
3.1.2 Tetragonal
The space group is p4. The Wyckoff positions and charges are shown in Table 3.1.
The charges are chosen to be the same as in the cubic structure.
Coordinate Charge
Sr (0, 0, 0) 2.0
Ti (0.5, 0.5, 0.5) 2.2
O (0.5, 0.5, 0) (0, 0.5, 0.5) -1.4
Table 3.1 Coordinates and charges for SrTiO3 in the tetragonal phase
According to x-ray diffraction measurements of the lattice parameters of
single-crystal SrTiO3 by F. W. Lytle [8], there is a small tetragonal distortion c/a=1.00056
below 110 K but the unit-cell volume remains unchanged through the transition. At 300 K,
a=3.905 Å. The volume is a3= 59.54744 Å3. In the tetragonal phase, the volume is 22
23
a2*c=59.54744 Å3.
Using the relation c/a=1.00056, we can find a=3.90427 Å and c=3.90646 Å. Therefore,
the lattice constants are chosen to be a=b=3.904 Å and c=3.906 Å.
3.1.3 Force field
Universal Force field 1.02 [29] was used. Sr6+2, Ti3+4 and O_2 were set as the types
for Sr, Ti and O, respectively. A five-character mnemonic label is used to describe the
atom types. The first two characters correspond to the chemical symbol; an underscore
appears in the second column if the symbol has one letter. The third column describes the
hybridization or geometry: 1 =linear, 2 = trigonal, R =resonant, 3 = tetrahedral, 4 =
square planar, 5 = trigonal bipyramidal, 6 = octahedral. The forth and fifth columns are
used as indicators of alternate parameters such as formal oxidation state. [29] Thus Sr6+2
indicates an octahedral Sr in the +2 oxidation state. The atomic data are shown in Table
3.2.
Table 3.2 Atomic data for Sr, Ti and O [29]
Atom
type Bond(Å)
Angle
(Degree)
Distance
(Å)
Energy
(kcal/mol)scale
Effective
charge
Sr6+2 2.052 90.0 3.641 0.235 12.0 2.449
Ti3+4 1.412 109.47 3.175 0.017 12.0 2.659
O_2 0.634 120.0 3.500 0.060 14.085 2.300
3.2 Energy minimization
Two different methods were used to determine the energy minimum. (1) In the first
method, the minimization algorithm of cerius2 searches the lattice constant and atomic
positions for the local minimum energy. (2) In the second method, the lattice constants
were changed systematically and the total potential energy for each lattice constant was
calculated. Thus we obtain a potential energy vs lattice constant curve. This allows us to
find the lattice constant from the condition that the energy is at a minimum.
3.2.1 One unit cell and its superlattice
Initially, the cubic structure which is shown in Figure 3.1 was used. Keeping the
fractional coordinates (relative distance and orientation) fixed, the current energy is
calculated for every given lattice constant. (see Figure 3.2) From the graph, we find that
when a = 3.619 Å, the energy is the minimum. The first minimization method also
confirms the result of 3.619 Å.
-2500.00
-2000.00
-1500.00
-1000.00
-500.00
0.003.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10
Lattice constant/ Å
Ener
gy/ c
al
Figure 3.2 Total potential energy vs lattice constant for SrTiO3
The simulation result from Katsumata et al [29] is a = 3.9056 Å, b = 3.9054 Å, c =
3.9055 Å. The difference between our result and their result may due to the different 24
25
simulation temperatures. Their calculation was fixed at 300 K, while our energy
minimization program is based on 0 K. Our result is also 7.3 % less than the experimental
data of 3.9051 Å at 300 K.
The superlattice was created from one unit cell. After creation of the superlattice, the
symmetry changes to p1. In the first minimization method, we allow all atoms to move
while keeping the fractional coordinates fixed. Not only the lattice constant a, b, and c,
but also the angle α, β and γ will change after minimization. The final structure gives that
a = b = c = 3.618 Å
α = β = γ =90.01o
Sr atoms are still located at the vertex of the cubic cell, Ti atoms are located at the
body center as before, but O atoms are no longer at the face center positions.
3.2.2 5*5*5 cell and its superlattice
The same procedure as in 3.2.1 was performed except the 5*5*5 cell was used for
the calculation. The simulation cell contains 625 particles. It turns out that the
energy-lattice constant graph is the same as Figure 3.2.
Another superlattice was built based on the 5*5*5 cell. The final lattice constant
from the first minimization method is 18.094 Å, which should be divided by 5 to get the
lattice constant for one unit cell. It gives that a = b = c = 3.619 Å, α = β = γ =90o.
To confirm the result, the second minimization method was performed. The
energy-lattice constant graph is shown in Figure 3.3. This gives the same lattice constant
3.619 Å.
Figure 3.3 Total potential energy vs lattice constant for 5*5*5 superlattice
3.3 Dynamics simulations
The tetragonal structure was used as the initial structure. The simulation was based
on the 5*5*5 supercell. The initial lattice constants are chosen to be a = b = 3.904 Å and
c = 3.906 Å. (see section 3.1.2) The temperature ranges from 60 K to 300 K. Constant
NVT was chosen at 60 K, and then constant NPT were chosen for the other temperatures.
The temperature was fixed for every run, while the structure was allowed to change to
reach the equilibrium at that temperature. T_Damping Thermostat was chosen to keep the
temperature constant. The relaxation time is 0.1 ps and the dynamics time step is 0.001 ps.
Number of steps was chosen large enough to make the system reach equilibrium.
The transition temperature is found to be around 130 K. The average value is 122.5
K, which is 16.7 % higher than the experimental value of 105 K [1].
26
CHAPTER IV
FIRST-PRINCIPLES CALCULATIONS ON SrTiO3
4.1 Structure of SrTiO3
Accelrys MS Modeling 4.0 is applied to build the structure. MS Modeling is a
flexible client-server software environment that provides advanced materials simulation
and modeling technology.
Figure 4.1 The structure of SrTiO3 built by MS Modeling
The space group of STO is pm3m. The Wyckoff positions are Sr (0, 0, 0), Ti (0.5,
0.5, 0.5) and O (0.5, 0.5, 0). The charges are chosen to be +2, +4, -2 for Sr, Ti and O,
respectively. The charges are used differently from those in molecular dynamics
simulation. Normal charges are always used in the ab initio calculations. The crystal
structure is shown in Figure 4.1.
27
28
4.2 Cambridge Sequential Total Energy Package
The Cambridge Sequential Total Energy Package (CASTEP) [31] is a program that
employs DFT to simulate the properties of solids, interfaces, and surfaces for a wide
range of materials classes. Based on total energy plane-wave pseudopotential methods,
CASTEP takes the number and types of atoms in a system and predicts properties
including lattice constant, molecular geometry, structural properties, band structures,
density of states, charge densities and wave functions, and optical properties.
4.3 Electronic structures and optical properties of SrTiO3
The crystal structure of STO shown in Figure 4.1 is loaded. Computations are
performed via the LDA with ultra-soft pseudopotentials. The initial lattice constant is
chosen to be the experimental value of 3.905 Å. [33] Geometry optimization is performed
at a plane-wave cutoff energy of 340 eV, with a 1×1×1 Monkhorst-Pack k-point mesh.
The relaxations are performed using the BFGS [34] algorithm while the fractional
coordinates are kept fixed during changes to the lattice.
The optimized lattice constant is 3.867 Å, slightly less than the value of 3.905 Å by
less than 1 %, as is typical in LDA calculations.
4.3.1 Band structure of SrTiO3
The calculated band structure of STO is shown in Figure 4.2. The G point in Figure
4.2 represents the Gamma point, Γ. Our band structure is similar to that in Samantaray et
al.’s work.[32]
Figure 4.2 The calculated energy band structure of SrTiO3
The valence band maximum is located at the R point, i.e., at the corner of the cubic
Brillouin zone. The lowest conduction band state is located at the zone center Γ, which
introduces an indirect band gap. There is a second local valence band maximum at Γ,
which gives rise to a direct band edge at Γ. The calculated direct band gap at the Γ point is
1.92 eV in Samantaray et al.’s work. [32] From Figure 4.2, our direct band gap at the Γ
point is 2.3 eV, which is closer to the experimental value of 3.2 eV. [35] But both
theoretical band gaps are smaller than experimental result. As is well known, the LDA
method underestimates the band gap in both semiconductors and insulators.
The band structure of STO has energetically separated low lying bands, which are
derived from the O 2s states. These bands are separated from a group of relatively narrow
bands arising from the Sr 4p states. Near the Fermi level in the valence band region, there
is a manifold of several bands, nominally derived from the O 2p states. The conduction
29
bands near the Fermi level have a strong Ti 3d character.
4.3.2 Total density of states of SrTiO3
The total density of states of STO is shown in Figure 4.3. We have obtained a similar
picture for the total density of states of STO as in Samantaray et al.’s work.[32]
Figure 4.3 Total density of states of SrTiO3
The lowest bands are located around -18 eV with double peaks and belong to O 2s
states. O 2s band states are -17.5 eV and Sr 4p band state is -15 eV. The Sr 4p state is
very close to O 2s states, which is in good agreement with the XPS measurement. [36, 37]
The upper valence band is made up predominately of the O 2p components. The width of
the O 2p valence band is 5.5 eV, close to 5 eV in Zollner et al.’s work. [38] The peak
nearest to the valence band maximum originates from the O 2p nonbonding components,
while the other two peaks correspond to the O 2p antibonding combinations of
oxygen-oxygen interactions. The lowest conduction bands are made up of Ti 3d bands.
30
The Ti 3d contribution is zero at the valence band maximum but rises strongly with
increasing binding energy, while the O 2p contribution rises from zero at the conduction
band minimum with increasing energy. These reflect the Ti 3d to O 2p covalency, i.e. the
O 2p orbitals are hybridized with Ti 3d orbitals. Following Ti 3d above the conduction
band, Sr 4d and O 3p contribute to the high energy region.
4.3.3 Charge density of SrTiO3
(a) (b)
Figure 4.4 Charge densities of SrTiO3 (atoms are colored as in Figure 4.1) (a) Sr-O plane (b) Ti-O plane
Figure 4.4 (a) and (b) display the calculated charge densities of STO on the Sr-O and
Ti-O planes. The high charge density region is around the nuclear sites and is nearly
spherically symmetric. There is very weak chemical bonding between Sr and O atoms,
which reflects the presence of an ionic bond between Sr and O atoms. From the density
of states of STO, it has been shown that there exists a significant hybridization of Ti 3d 31
with O 2p states. This means that the bonding in Ti-O system cannot be purely ionic but
must exhibit a large covalent part. Figure 4.4 (b) presents a clearer picture about the
nature of chemical bonding between Ti and O atoms. It confirms that the Ti-O bond has a
strong covalent character. This is apparent from the noticeable overlapping charge
distribution at the middle of the Ti-O bond. However there is not much bonding charge to
link the Sr and O atoms, it indicates that the bonding between Sr and O atoms is mainly
ionic.
4.3.4 Optical properties of SrTiO3
The optical properties can be derived from the complex dielectric function
1 2( ) ( ) ( )iε ω ε ω ε ω= + , where 1( )ε ω and 2 ( )ε ω are the real part and the imaginary part
of the dielectric constant, respectively. The refractive ( )n ω and extinction coefficient
( )k ω follow from Equations 4.1 and 4.2.
2 21 2 1
1( ) ( )[ ( ) ( ) ( )]2
n ω ε ω ε ω ε= + + 1/ 2ω (4.1)
2 21 2 1
1( ) ( )[ ( ) ( ) ( )]2
k ω ε ω ε ω ε= + − 1/ 2ω (4.2)
Figure 4.5 shows the calculated optical properties of STO. Comparing our result
with Samantaray et al.’s work [32], we find that we have obtained similar results.
32
Figure 4.5 Refractive index and extinction coefficient of SrTiO3
4.4 Summary of the work on SrTiO3
The optimized lattice constant a=3.619 Å and a=3.867 Å from molecular dynamics
simulations and ab initio calculations, respectively. The result from ab initio calculations
is in good agreement with the experimental value of 3.905 Å, based on the consideration
that underestimation is the typical limitation is LDA calculations. The difference between
the result from molecular dynamics simulations and the experimental value is due to the
different temperatures. The experimental value is measured at 300 K while our
minimization result is calculated at zero temperature.
The transition temperature is found to be 122.5 K, which is 16.7 % higher than the
experimental value of 105 K. Molecular dynamics simulations are not sufficient to find
the structural transition temperatures.
33
34
The calculated electronic properties show that there is a strong covalent bonding
between Ti and O, which explains why TiO6 octahedron behaves as a rigid body
throughout the phase transition.
CHAPTER V
FIRST-PRINCIPLES CALCULATIONS ON CaCu3Ti4O12
5.1 Structure of CaCu3Ti4O12
Accelrys MS Modeling 4.0 is applied to build the structure. The space group of
CCTO is Im3. The Wyckoff positions are shown in Table 5.1. The charges are chosen to
be +2, +2, +4, -2 for Ca, Cu, Ti and O, respectively. The crystal structure of CCTO is
shown in Figure 5.1.
Table 5.1 The Wyckoff positions for CaCu3Ti4O12
Atom Position
Ca (0, 0, 0)
Cu (1/2, 0, 0)
Ti (1/4, 1/4, 1/4)
O (0.303, 0.175, 0)*
*The structural parameters for the O atom are those of Ref [20]
35
Figure 5.1 The structure of CaCu3Ti4O12 built by MS Modeling
This structure can be obtained from the ideal structure of CaTiO3 by replacing three
out of every four Ca site ions with a Cu ion, which quadruples the cell to CaCu3Ti4O12,
and then performing a correlated rotation of the four octahedron until the Cu ion is
fourfold coordinated with O ions in a nearly square arrangement-the bonds connecting Cu
and O are all the same length, though the angles between bonds deviate slightly from 90
degrees. All the following calculations are performed using this 40-atom cell in Figure
5.1.
5.2 Energy minimization of CaCu3Ti4O12
The crystal structure of CCTO shown in Figure 5.1 is loaded. Computations are
performed via the LDA with ultra-soft pseudopotentials. The highest occupied p shell
36
electrons, for Ca and Ti, are treated as valence. The potential considers electrons in the 3d
and 4s shells of Cu as valence electrons. Energy minimization is performed at a
plane-wave cutoff energy of 500 eV (equivalent to 37 Ry [20]), with a 2×2×2
Monkhorst-Pack k-point mesh.
The total energy is computed while the lattice constant and the fractional coordinates
are kept fixed. The lattice constant is changed in steps and each time the energy is
determined. The total energy vs lattice constant is shown in Figure 5.2. Therefore, we
find the equilibrium lattice constant when the total energy is minimum.
-33450
-33440
-33430
-33420
-33410
-33400
-333906.3 6.5 6.7 6.9 7.1 7.3 7.5 7.7 7.9 8.1 8.3 8.5
Lattice constant/ Å
Tota
l ene
rgy/
eV
Figure 5.2 Energy minimization of CaCu3Ti4O12
The lattice constant ranges from 6.50 Å to 8.10 Å. a = 7.29 Å is found to be the
lattice constant for the minimum energy. It is the same result as in He et al.’s work. [20]
The experimental value is 7.384 Å measured at 35 K [20], which is 1 % more than our
LDA result. This is the common limitation in LDA calculation.
37
5.3 Geometry Optimization of CaCu3Ti4O12
The crystal structure of CCTO shown in Figure 5.1 is loaded. Computations are
performed via the LDA with ultrasoft pseudopotentials. The initial lattice constant is
chosen to be our calculated lattice constant 7.29 Å. Geometry optimization is performed
at a plane-wave cutoff energy of 500 eV which results in convergence of the total energy
to 1 meV/atom, with a 2×2×2 Monkhorst-Pack k-point mesh. The relaxations are
performed using the BFGS [34] algorithm. The internal parameters of the structure are
relaxed at our calculated lattice constant.
The optimized lattice constant is 7.29 Å, which confirms the result from energy
minimization. The Wyckoff position for O is (0.305, 0.174, 0), which is somewhat
different from that in He et al.’s work. [20] while all the other positions for Ca, Cu and Ti
are the same.
Figure 5.3 Bond lengths and angles of CaCu3Ti4O12 after Geometry Optimization (bond lengths are in blue and angles are in black.)
38
Bond lengths and angles after geometry optimization are shown in Figure 5.3. The
Cu cation along with its four nearest neighboring O atoms forms a slightly distorted
square of CuO4 with Cu-O distance of 1.907 Å, 0.68% smaller than 1.92 Å in He et al.’s
work [20] and O-Cu-O angle of 96.498o. The four next nearest neighboring oxygens are
2.785 Å away from Cu, with O-Cu-O angle of 118.153o. The third nearest neighboring
oxygens are 3.252 Å away from Cu, with O-Cu-O angle of 93.808o. These three kinds of
CuO4 complexes are orthogonal to one another.
5.4 Band structure of CaCu3Ti4O12
Figure 5.4 Band structure of CaCu3Ti4O12
Figure 5.4 shows the calculated band structure of CCTO. The unoccupied
conduction bands above 1 eV have predominant Ti 3d character. The bands below 1 eV
are composed of the Cu 3d-O 2p manifold. The Fermi energy is 0.48 eV. The direct band 39
gap is 0.57 eV, which is greater than 0.27 eV in He et al.’s work [20] and closer to the
experimental optical gap of 1.5 eV as a lower limit.[see ref. 20 in (20)] The underestimation of
the gap is common in LDA calculations and results from the underestimation of
correlation effects.
5.5 Density of states of CaCu3Ti4O12
Figure 5.5 Density of states of CaCu3Ti4O12(Red line indicates the Fermi level.)
Our density of states shown in Figure 5.5 is similar to that presented by He et al.[20]
The bands shown in the energy range in Figure 5.4 are mainly O 2p and Cu 3d orbitals.
The bands from -7.25 eV to -0.5 eV consist mainly of Cu 3d orbitals that hybridize
weakly with O 2p orbitals that point toward their Ti neighbors. In the range from -0.5 eV
to 3 eV around the Fermi level, the band structure reflects two narrow bands each of
40
which contains three bands. These six isolated bands consist of antibonding interactions
of Cu 3d orbitals and O 2p orbitals pointing to the Cu from its four near oxygen
neighbors. These are responsible for the magnetism of the compound.
5.6 Charge densities of CaCu3Ti4O12
The total charge density is symmetric and the spin density is antisymmetric, under
the fractional lattice translation (½, ½, ½), consistent with an AFM state. Spin-up and
spin-down charge densities are shown in Figure 5.6. It indicates antibonding interactions
between Cu 3d and O 2p, which extend over the central cluster composed of a Cu ion and
its four nearest O neighbors.
Figure 5.6 Spin-up (left panel) and spin-down (right panel) charge densities of CaCu3Ti4O12
41
42
CHAPTER VI
FIRST-PRINCIPLES CALCULATIONS ON Ca1+xCu3-xTi4O12
We performed first-principles calculations on Ca1+xCu3-xTi4O12 (x= -1, 1, 2, 3).
Different Ca and Cu ratios give different materials, i.e. CuTiO3, Ca2Cu2Ti4O12,
Ca3CuTi4O12, CaTiO3. Electronic properties are calculated.
6.1 Structure
Accelrys MS Modeling 4.0 is applied to build the structure. The structures are
shown in Figure 6.1-6.4. The structures of CuTiO3, Ca3CuTi4O12 and CaTiO3 are built
based on the structure of CaCu3Ti4O12 shown in Figure 5.1. CuTiO3 is built by
substituting Ca by Cu. The crystal structure of CuTiO3 is shown in Figure 6.1.
Ca3CuTi4O12 is built by exchanging the Ca and Cu positions. The crystal structure of
Ca3CuTi4O12 is shown in Figure 6.3. CaTiO3 is built by substituting Cu by Ca. The crystal
structure of CaTiO3 is shown in Figure 6.4.
However, the structure of Ca2Cu2Ti4O12 can be built from the structure of
CaCu3Ti4O12, so this structure is built separately from the others. The space group of
Ca2Cu2Ti4O12 is Pm3. The Wyckoff positions are shown in Table 6.1. The charges are
chosen to be +2, +2, +4, -2 for Ca, Cu, Ti and O, respectively. The crystal structure of
Ca2Cu2Ti4O12 is shown in Figure 6.2.
43
Figure 6.2 The structure of Ca2Cu2Ti4O12 Figure 6.1 The structure of CuTiO3
Figure 6.3 The structure of Ca3CuTi4O12 Figure 6.4 The structure of CaTiO3
44
Table 6.1 The Wyckoff positions for Ca2Cu2Ti4O12
Atom Position
Ca (0, 0, 0) (1/2, 1/2, 0)
Cu (1/2, 0, 0) (1/2, 1/2, 1/2)
Ti (1/4, 1/4, 1/4)
O (1/4, 1/4, 0) (1/4, 1/4, 1/2)
6.2 Geometry optimization
Computations are performed via the LDA with ultra-soft pseudopotentials. The
initial lattice constant is 7.29 Å which is the equilibrium lattice constant from our former
single energy calculation for CCTO. Geometry optimization is performed at a plane-wave
cutoff energy of 500 eV, with a 2×2×2 Monkhorst-Pack k-point mesh. The relaxations are
performed using the BFGS [34] algorithm and all the atoms are allowed to relax during
changes to the lattice.
For CuTiO3, the optimized lattice constant is found to be 7.25 Å. The optimization
result shows that Cu and Ti atoms are still in the original positions, which agrees well
with the fact that the CuTiO3 structure is perovskite structure. The O atoms move from
their original positions, but the result doesn’t show obvious tendency of moving towards
to the center of each side of the primitive cell CuTiO3.
For Ca2Cu2Ti4O12, the optimized lattice constant is found to be 7.58 Å.
45
For Ca3CuTi4O12, the optimized lattice constant is found to be 7.55 Å. From the
optimized structure, Ca, Cu and Ti atoms are still in the original positions. The O atoms
move towards the center of each side of the primitive cell, which shows a tendency
towards pure perovskite structure. Ca3CuTi4O12 is a composite of CaCu3Ti4O12 and
CaTiO3. (CaCu3Ti4O12 : CaTiO3=1:8) Since it is rich in CaTiO3, its properties resemble
CaTiO3 which is a perovskite structure.
For CaTiO3, the optimized lattice constant is found to be 7.57 Å. Wang et al. [39]
performed generalized gradient approximation (GGA) calculations using the plane-wave
pseudopotential method to find that the theoretical lattice constant is 3.88 Å. Since we
use 40-atom cell in our calculations, the lattice constant is 7.57/ 2= 3.785 Å in a 20-atom
cell, which is 2.5 % smaller than the theoretical one in Wang et al.’s work. [39] Compared
with the experimental one of 3.8950 Å [40], our theoretical lattice constant is 2.8 % smaller.
The optimized structure shows that Ca and Ti atoms are still in the original positions. The
O atoms move towards to the center of each side of the primitive cell CaTiO3. CaTiO3 has
perovskite structure. Therefore, during optimization the structure changes from that of
CaCu3Ti4O12 structure to that of CaTiO3 as we expected.
The summary of optimized lattice constants for different Ca, Cu ratio is shown in
Table 6.2. As the Ca, Cu ratio increases from 0 to ∞, the optimized lattice constant
increases as well except for Ca2Cu2Ti4O12, which shows a divergence. As we know, the
pauling ionic radii of Ca2+ is 99 pauling radius/pm and that of Cu2+ is 96 pauling
radius/pm. Since Ca2+ > Cu2+, the lattice constant should increase if more calcium is
added to the material. This agrees with our result.
As for the abnormal behavior of Ca2Cu2Ti4O12, one reason may be the simulation
46
structure of Ca2Cu2Ti4O12 is not based on the structure of CCTO, while the structures of
the others are all based on the structure of CCTO.
Table 6.2 Summary of optimized lattice constant for different Ca, Cu ratio
Material type Ca, Cu ratio Optimized lattice constant
CuTiO3 Ca: Cu=0 7.25 Å
CaCu3Ti4O12 Ca: Cu=1:3 7.29 Å
Ca2Cu2Ti4O12 Ca: Cu=2:2 (1:1) 7.58 Å
Ca3CuTi4O12 Ca: Cu=3:1 7.55 Å
CaTiO3 Ca: Cu=∞ 7.57 Å
6.3 Band structure
The band structures of CuTiO3, Ca2Cu2Ti4O12, Ca3CuTi4O12, CaTiO3 are shown in
Figure 6.5-6.8.
For CuTiO3, the direct band gap at the Γ point is 0.53 eV. The indirect band gap,
which is between the valence band maximum Γ point and the conduction band minimum
X point, is 0.65 eV. The Fermi level is 0.32 eV.
For Ca2Cu2Ti4O12, the direct band gap at the Γ point is 1.22 eV. The indirect band
gap, which is between the valence band maximum Γ point and the conduction band
minimum X point, is 1.99 eV. The Fermi energy is 0.19 eV.
For Ca3CuTi4O12, the direct band gap at the Γ point is 2.30 eV. The indirect band gap,
which is between the valence band maximum Γ point and the conduction band minimum
X point, is 2.36 eV. The Fermi energy is 0.19 eV.
For CaTiO3, the direct band gap at the Γ point is 2.20 eV. The indirect band gap,
which is between the valence band maximum Γ point and the conduction band minimum
X point, is 2.30 eV. The Fermi energy is 0.27 eV.
47
Figure 6.5 Band structure of CuTiO3
Figure 6.7 Band structure of Ca3CuTi4O12
Figure 6.6 Band structure of Ca2Cu2Ti4O12
Figure 6.8 Band structure of CaTiO3
48
The direct band gaps are summarized in Table 6.3. With the increase of Ca in the
material, the band gap increases. The increasing band gap indicates the material is more
like an insulator. Therefore, with the increase of Ca in the material, insulator character
becomes much more pronounced. This is in good agreement with the experimental
results.
Table 6.3 Summary of direct band gap for different Ca, Cu ratio
Material type Direct band gap
CuTiO3 0.53 eV
CaCu3Ti4O12 0.57 eV
Ca2Cu2Ti4O12 1.22 eV
Ca3CuTi4O12 2.30 eV
CaTiO3 2.20 eV
6.4 Density of states
The densities of states of CuTiO3, Ca2Cu2Ti4O12, Ca3CuTi4O12, CaTiO3 are shown in
Figure 6.9-6.12. The dashed line does not indicate the Fermi level in every figure. The
Fermi level is located at 0.32 eV, 0.19 eV, 0.19 eV and 0.27 eV for CuTiO3, Ca2Cu2Ti4O12,
Ca3CuTi4O12, CaTiO3, respectively.
The bands below the valence band maximum are mainly O 2p and Cu 3d in
character. Cu 3d orbitals hybridize weakly with O 2p orbitals. The lowest conduction
bands are made up of Ti 3d bands, which hybridize with O 2p orbitals. Ca contribution is
quite small in this energy range.
Figure 6.9 Density of states of CuTiO3
Figure 6.10 Density of states of Ca2Cu2Ti4O12
49
Figure 6.11 Density of states of Ca3CuTi4O12
Figure 6.12 Density of states of CaTiO3
6.5 Charge density
The charge densities of CuTiO3, Ca2Cu2Ti4O12, Ca3CuTi4O12, CaTiO3 are calculated.
They show similar results. In this thesis, the charge densities of Ca2Cu2Ti4O12 are shown
as an example. Figure 6.13 (a) and (b) shows the charge densities of Ca2Cu2Ti4O12 on the
50
Ca-Cu-O and Ti-O planes. The high charge density region is around the nuclear sites and
is nearly spherically symmetric. The Ti-O bond has a strong covalent character which
results from the noticeable overlapping charge distribution at the middle of the Ti-O bond.
Figure 6.13 Charge densities of Ca
There is no chemical bonding between any other types of atoms, e.g., Ca-O or Cu-O.
.6 Summary of the work on Ca1+xCu3-xTi4O12
CCTO are in good agreement with other
ind that with increase of Ca in the material, the optimized
2Cu2Ti4O12(atoms are colored as in Figure 6.2)
(b) (a)
(a) Ca-Cu-O (b) Ti-O
6
Our calculated ground state properties of
people’s work.[20] The optimized lattice constant is 7.29 Å, which is 1 % less than the
experimental value of 7.384 Å. The density of states and charge densities confirm the
AFM character of CCTO.
For Ca1+xCu3-xTi4O12, we f
51
52
lattice constant and band gap increase, and the insulator character becomes much more
pronounced in the material. This is in good agreement with the experimental results.
53
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