Download Materials Design from ab initio Calculations

Comprehensive Summaries of Uppsala Dissertations
from the Faculty of Science and Technology 982
Materials Design from ab initio
Calculations
BY
SA LI
ACTA UNIVERSITATIS UPSALIENSIS
UPPSALA 2004
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Printed in Sweden by Universitetstryckeriet, Uppsala 2004
To my family
Front page illustration
The electronic charge density distribution for
the -AlOOH phase in the (001) plane.
List of Publications
I S. Li, R. Ahuja, and B. Johansson
Pressure-induced phase transitions of KNbO 3
J. Phys. C 14, 10873 (2002)
II Sa Li, R. Ahuja, and B. Johansson
High pressure theoretical studies of actinide dioxides
High Pressure Research 22, 471 (2002)
III J. K. Dewhurst, R. Ahuja, S. Li, and B. Johansson
Lattice Dynamics of Solid Xenon under Pressure
Phys. Rev. Lett. 88, 075504 (2002)
IV B. Holm, R. Ahuja, S. Li, and B. Johansson
Theory of the ternary layered system Ti-Al-N
J. of Appl. Phys. 91, 9874 (2002)
V H.W.Hugosson, G.E.Grechnev, R.Ahuja, U.Helmersson, L.Sa,
and O.Eriksson
Stabilization of potential superhard RuO 2 phases: A
theoretical study
Phys. Rev. B 66, 174111 (2002)
VI Z. M. Sun, R. Ahuja, S. Li, and J. M. Schneider
Structure and bulk modulus of M2 AlC
Appl. Phys. Lett. 83, 899 (2003)
VII S. Li, R. Ahuja and Y. Wang, and B. Johansson
Crystallographic structures of PbWO 4
High Pressure Research 23, 343 (2003)
VIII Z. M. Sun, S. Li, R. Ahuja, and J. M. Schneider
Calculated elastic properties of M2 AlC
Solid State Commun. 129, 589 (2004)
IX A. B. Belonoshko, S. Li, R. Ahuja, and B. Johansson
High-pressure crystal structure studies of Fe, Ru and
Os
J. Phys. Chem. Sol. (In press)
X S. Li, R. Ahuja, and B. Johansson
Wolframite: the post-fergusonite phase in YLiF 4
J. Phys. C 16, 983 (2004)
XI S. Li, R. Ahuja, and B. Johansson
The Elastic and Optical Properties of the High-Pressure
Hydrous Phase δ-AlOOH
Submitted to Phys. Rev. B
XII J. -P. Palmquist, S. Li, P. O. A. Persson, J. Emmerlich, O.
Wilhelmsson, H. Högberg, M. I. Katsnelson, B. Johansson,
R. Ahuja, O. Eriksson, L. Hultman, and U. Jansson
New MAX Phases in the Ti-Si-C System Studied by
Thin Film Synthesis and ab initio Calculations
Submitted to Phys. Rev. B
XIII P. Finkel, J. D. Hettinger, S. E. Lofland, K. Harrell, A. Ganguly, M. W. Barsoum, Z. sun, S. Li, and R. Ahuja
Low Temperature Elastic, Electronic and Transport
Properties of Ti3 Six Ge1−x C2 Solid Soutions
Submitted to Phys. Rev. B
XIV M. Magnuson, J. -P. Palmquist, M. Mattesini, S. Li, R.
Ahuja, O. Eriksson, J.Emmerlich, O. Wilhelmsson, P. Eklund, H. Högberg, L. Hultman, and U. Jansson
Electronic structure of the MAX-phases Ti 3 AC2 (A=Al,
Si, Ge) investigated by soft X-ray absorption and
emission spectroscopies
Submitted to Phys. Rev. B
XV Z. M. Sun, D. Music, R. Ahuja, S. Li, and J. M. Schneider
Bonding and classification of nanolayered ternary carbides
Submitted to Phys. Rev. B
XVI R. Ahuja, H. W. Hugosson, S. Li, B. Johansson, and O.
Eriksson
Electronic structure and optical properties of C 60
Submitted to Phys. Rev. B
XVII R. Ahuja, L. M. Huang, S. Li, and Y. Wang
High pressure structural phase transition in Zircon
(ZrSiO4 )
Submitted to Phys. Rev. B
XVIII A. Grechnev, S. Li, R. Ahuja, O. Eriksson, and O. Jansson
A possible MAX new phase, Nb3 SiC2 , predicted from
First Principles Theory
Submitted to Appl. Phys. Lett.
XIX S. Li, H. Pettersson, C. G. Ribbing, B. Johansson, M. W.
Barsoum, and R. Ahuja
Optical properties of Ti3 SiC2 and Ti4 AlN3
In manuscript
Comments on my contribution to the papers
For those papers I, II, VII, X, XI and XIX, I’m the first author of, I
took the major responsibilities in calculation design and their execution,
as well as paper drafting. For papers IX and XIV, I have performed calculations and contributed to the paper writings. In the experimental
paper XII, I have carried out calculations and written up the theoretical parts of the publication. As regards paper III, IV, V, VI, VIII, IX,
XIII, XV, XVI, XVII and XVIII, I have contributed either with ideas
or calculations and result analysis.
Contents
1 Introduction
2 Many body problem
2.1 Introduction . . . . . . . . .
2.2 The Hartree approximation
2.3 Hartree-Fock approximation
2.4 Density functional theory .
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3 Computational methods
3.1 Introduction . . . . . . . . . . . . . .
3.2 Electronic structure methods . . . .
3.3 The LMTO method . . . . . . . . .
3.3.1 Muffin-tin orbitals . . . . . .
3.3.2 The LMTO-ASA method . .
3.4 Full potential LMTO method . . . .
3.4.1 The basis set . . . . . . . . .
3.4.2 The LMTO matrix . . . . . .
3.4.3 Total energy . . . . . . . . .
3.5 Projector Augmented Wave Method
3.5.1 Wave function . . . . . . . .
3.5.2 Charge density . . . . . . . .
3.5.3 Total energy . . . . . . . . .
3.6 Ultrasoft pseudopotential . . . . . .
3.7 PAW and US-PP . . . . . . . . . . .
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4 Phase transitions
4.1 Static total energy calculation . . . . . . .
4.2 Elastic stability criteria . . . . . . . . . .
4.3 Bain path . . . . . . . . . . . . . . . . . .
4.4 Dynamical stability (phonon calculation) .
4.5 Equation of state . . . . . . . . . . . . . .
4.5.1 Murnaghan equation of state . . .
4.5.2 Birch-Murnaghan equation of state
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4.5.3
4.5.4
Universal equation of state . . . . . . . . . . . . . 31
Comparison of different EOS for Fe . . . . . . . . . 31
5 Semiconductor optics
33
5.1 Dielectric function . . . . . . . . . . . . . . . . . . . . . . 33
5.2 Dielectric function of PbWO4 . . . . . . . . . . . . . . . . 34
5.3 Solar energy materials . . . . . . . . . . . . . . . . . . . . 36
6 MAX phases
6.1 MN+1 AXN phase . . . . . . . . .
6.2 Phase stability in Ti-Si-C system
6.3 Chemical bonding in 312 phases
6.4 XAS and XES calculation . . . .
6.5 DOS with electrical conductivity
6.6 Optical properties . . . . . . . .
6.7 Surface energy . . . . . . . . . .
6.8 Ductility . . . . . . . . . . . . . .
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7 Sammanfattning på svenska
53
Acknowledgments
57
Bibliography
59
Chapter 1
Introduction
In 1998, Walter Kohn was awarded Nobel Prize in chemistry. His
work has enabled physicists and chemists to calculate the properties of
molecules and solids using computers, without performing experiments
in the laboratory. Already around 1930 physicists were fully aware of
the quantum mechanical equations governing the behavior of systems of
many electrons, but were incapable of exactly solving them in all but the
very simplest cases. They developed several approximation schemes, but
none of them was very successful. In 1964 Walter Kohn [1] and Pierre
Hohenberg and somewhat later Walter Kohn and Lu Jeu Sham [2] had
proved an idea that was essential to their solution scheme, which is
now called density functional theory (DFT). DFT differs from quantum
chemical methods and does not yield a correlated N-body wavefunction.
Many of the chemical and electronic properties of molecules and solids
are determined by electrons interacting with each other and with atomic
nuclei. In DFT, knowing the average density of electrons at all points
in space is enough to uniquely determine the total energy, hence also
a number of other properties of the system. DFT theory is based on
one-electron theory and shares many similarities with the Hartree-Fock
method. DFT has come to prominence over the last decade as a method
potentially capable of generating very accurate results at relatively low
cost.
By means of state-of-the-art DFT, great achievements in calculating
mechanical and electronic properties of solids have been made. Different
approximations related to the construction of exchange-correlation functionals used in DFT for ground and excited states have been introduced,
for example, local density approximation (LDA) [3] and generalized gradient approximations (GGA) [4], LDA+U [5] and GW [6] approximation.
Applications to materials modeling in general as well as to nanotubes,
quantum dots, and artificial molecules are incorporated. The calcula3
4
CHAPTER 1. INTRODUCTION
tions have developed so rapidly and reached such an advanced level that
it now becomes the right time for a theory-based approach, to support
as well as supplement experiment. This development has also been facilitated by the continued upgrading of powerful computers, which have
made it possible to calculate materials properties with an impressive accuracy. The main thrust of the present thesis is to use highly accurate
theoretical ab initio methods to predict materials properties and search
for the new engineering materials.
Recently, the nanolayered ternary compounds M N+1 AXN (MAX)
[7], where N =1, 2 or 3, M is an early transition metal, A is an Agroup (mostly IIIA and IVA) element, and X is either C and/or N, have
attracted increasing interests owing to their unique properties. These
ternary carbides and nitrides combine the unique properties of both
metals and ceramics. Like metals, they are good thermal and electrical conductors with electrical and thermal conductivities ranging from
0.5 to 14×106 Ω−1 m−1 [8], and from 10 to 40 W/m·K [9], respectively.
They are relatively soft with Vickers hardness of about 2-5 GPa. Like
ceramics, they are elastically stiff, some of them, like Ti 3 SiC2 , Ti3 AlC2
and Ti4 AlN3 also exhibit excellent mechanical properties at high temperatures. They are resistant to thermal shock and unusually damage
tolerant, and exhibit excellent corrosion resistance. Above all, unlike
conventional carbides or nitrides, they can be machined by conventional
tools without lubricant. All these excellent properties make MAX phases
a new family of technologically important materials. Systematic studies
on MAX phases regarding their electronic, bonding, elastic and optical
properties have been carried out and described in this thesis. The relation between mechanical and bonding properties of these solids elaborated by the present electronic structure studies provides robust support
to the exploration of the behavior of these ternary compounds.
The rest of the thesis is organized as the following. The second
chapter describes the main idea of density functional theory. The third
chapter deals with the computational methods. My research results can
be classified in three parts, 1) Phase transition and its related properties,
such as phase stability and equations of states, which are elaborated in
Chapter 4. 2) The calculated results for the linear optical properties of
certain semiconductors, such as the scintillating crystal PbWO 4 and the
solar energy materials CuIn1−x Gax Se2 are described in Chapter 5. 3)
In Chapter 6, applications of DFT in MAX phases are presented, which
include their electronic, bonding, mechanical and optical properties.
Chapter 2
Many body problem
2.1
Introduction
In this work our focus is restricted to bulk materials. This means that
possible surface effects are excluded and that we consider the bulk to
be an infinite crystal. To study the properties of atoms, molecules and
solids, the so-called Schrödinger Equation has become the basic tool that
the solid state theorists work with. The time-independent Schrödinger
Equation has the form
HΨ = EΨ,
(2.1)
which has been proven to quite exactly solve the problem with one
nucleus and one electron, such as the hydrogen atom. However for
a solid the system is described by the many electron wave function
Ψ(r1 , r2 , ..., rN ), where ri gives the position and spin of particle i. In
a solid we are typically dealing with 1023 particles and this makes the
problem very complex. Let us have a closer look at the Hamiltonian
that describes the whole bulk system:
H = −
+
h̄2 2
h̄2 2 1 Z 2 e2
−
i +
2 k 2Mk
2m i
2 k=l |Rk − Rl |
e2
Ze2
1
−
,
2 i=j |ri − rj |
|ri − Rk |
i
k
(2.2)
where h̄ is Planck constant, Rk is the nuclear coordinate for the k’th
nucleus, ri the electronic coordinate for the i’th electron and M k and
m are the corresponding masses. Z is the nuclear charge. The first two
terms in Eq. (2.2) are the kinetic energy operators for the nuclei and
electrons, respectively and the third term describes the nuclei-nuclei
interaction, VNN . The next term in Eq. (2.2) is the electron-electron
interaction, Vee . The last term is the interaction between the electrons
5
CHAPTER 2. MANY BODY PROBLEM
6
and nuclei and could be regarded as an external potential, V ext , acting
upon the electrons.
Since the Hamiltonian describes a strongly coupled system involving
both electrons and nuclei, it is quite difficult to solve. The first approximation we introduce is Born-Oppenheimer approximation. Within this
approximation the nuclei are taken to be stationary, so that the nuclear
kinetic energy will be zero. Now the total energy Hamiltonian can be
expressed as
H = Te + VNN + Vee + Vext = −
+
2.2
h̄2 2 1 Z 2 e2
i +
2m i
2 k=l |Rk − Rl |
e2
Ze2
1
−
2 i=j |ri − rj |
|ri − Rk |
i
k
(2.3)
The Hartree approximation
The Hartree approximation provides one way to reduce Eq. ( 2.3) to
a problem which we can solve easily. In Eq. (2.3), the potential which
a certain electron feels depends upon all the other electrons’ positions.
However this potential can be approximated by an average single-particle
potential
|ψj (rj )|2
nj
,
(2.4)
Vd (ri ) = e2
|ri − rj |
j=i
where nj are the orbital occupation numbers and ψ j (rj ) is a singleparticle wave-equation, i.e. a solution to the one-particle wave-equation,
h̄2 2
−
+Vext + Vd (ri ) ψi (ri ) = εi ψi (ri )
2m
(2.5)
With this simplification the set of equations now become separable.
However the equations are still non-linear and have to be solved selfconsistently by iteration.
According to the Pauli exclusion principle, two electrons can not
be in the same quantum state. However the wave function in Hartree
theory
Ψ(r1 σ1 , r2 σ2 ..., rN σN ) =
N
ψi (ri , σi )
(2.6)
i
is not antisymmetric under the interchange of electron coordinates and
accordingly does not follow the Pauli principle. Furthermore, the Hartree
approximation fails to represent how the configuration of the N − 1 electrons affects the remaining electrons. This problem has been rectified
by Hartree-Fock theory.
2.3. HARTREE-FOCK APPROXIMATION
2.3
7
Hartree-Fock approximation
We assert that a solution to HΨ = EΨ is given by any state Ψ that
makes the following quantity stationary:
E=
(Ψ, HΨ)
.
(Ψ, Ψ)
(2.7)
According to the variational principle [10], the normalized expectation
value of energy is minimized by the ground-state wave function Ψ.
A better description is to replace wave function Eq. (2.6) by a Slaterdeterminant of one-electron wave functions
ψ (r σ ) ψ (r σ ) · · ·
1 2 2
1 1 1
1 ψ2 (r1 σ1 ) ψ2 (r2 σ2 ) · · ·
Ψ(r1 σ1 , r2 σ2 ..., rN σN ) = √ ..
..
..
.
N! .
.
ψN (r1 σ1 ) ψN (r2 σ2 ) · · ·
.
ψN (rN σN ) ψ1 (rN σN )
ψ2 (rN σN )
..
.
(2.8)
This is a linear combination of products of the form given by of Eq.
(2.6) and all other products obtainable from the permutation of the r i σi
among themselves. The Hartree-Fock equation which follows from an
energy-minimization is given by:
−
j
2
h̄
2 +Vext (ri ) + Vd (ri ) ψi (ri )
− 2m
2
e
∗ dr |r−r
| ψj (r )ψi (r )ψj (r)δsi sj = εi ψi (ri ).
(2.9)
The last term on the left side due to exchange originates from the wave
function (Slater determinant). This term only operates between electrons having the same spin, this is called the exchange term. In addition
to this, there should also be a correlation interaction between electrons,
which is not included here. Consequently, the correlation energy can be
described as the difference between the exact energy and the HartreeFock energy. Another more effective approach to treat the electrons in
a solid will be introduced in the following sections.
2.4
Density functional theory
The density functional theory is based on two fundamental theorems
introduced by Hohenberg and Kohn [1], and later extended by Kohn
and Sham [2].
First, the ground-state energy E of a many electron system is shown
to be a unique functional of the electron density n(r),
E[n] =
drVext (r)n(r) + F [n].
(2.10)
CHAPTER 2. MANY BODY PROBLEM
8
According to the Hohenberg and Kohn theory, we can separate the functional F [n] into two terms
F [n] =
n(r)n(r )
drdr + G[n].
|r − r |
(2.11)
The first term on the right is the usual electron-electron Coulomb contribution, and the second term G[n] is a universal functional of the electron
density. Kohn and Sham proposed the following approximation for the
functional G[n],
(2.12)
G[n] = T [n] + Exc [n],
where T [n(r)] is the kinetic energy of a system of non-interacting electrons with electron density n(r). However, it is impossible to find an
exact expression for the exchange-correlation energy E xc .
To deal with the problem of the exchange-correlation energy E xc , a
most useful approximation has been introduced, namely the local density
approximation (LDA). This approximation is exact in the limit of slowly
varying densities. In LDA the exchange-correlation energy is replaced
by
Exc [n] =
n(r)εxc [n]dr,
(2.13)
where the εxc is the exchange and correlation energy per particle of a
homogeneous electron gas.
Now we write the electron density in terms of one-electron wavefunctions, ψ(r), as
n(r) =
N
ψi∗ (r)ψi (r),
(2.14)
i=1
where N is the total number of electrons.
The one-particle Schrödinger equation now becomes
[− 2 +Veff (r)]ψi (r) = εi ψi (r),
(2.15)
where the atomic unit h̄ = 2me = e2 /2 = 1 has been used. The effective
one-electron potential, Veff , is given by
Veff (r) = Vext (r) +
where
Vxc (r) =
2n(r )
dr + Vxc (r),
|r − r |
δ(n(r)εxc [n(r)])
.
δ(n(r))
(2.16)
(2.17)
The set of equations, (2.15) − (2.17) are known as the Kohn-Sham equations, which have to be solved in a self-consistent way, just like the
Hartree and Hartree-Fock approximations.
Chapter 3
Computational methods
3.1
Introduction
In the previous chapter we have obtained an effective one-electron equation which can be solved in a self-consistent way,
[− 2 +Veff (r)]ψi (r) = εi ψi (r).
(3.1)
The method of solving this eigenvalue equation makes use of the symmetry of the crystal structure.
For an infinite crystal the potential is periodic, i.e. invariant under
lattice translations R. For a monoatomic solid we have
V (r + R) = V (R),
(3.2)
R = n 1 a1 + n 2 a2 + n 3 a3 ,
(3.3)
where R is defined by
in which ni are integers and the set of vectors a i are the real space Bravais lattice vectors that span the crystal cell. According to the Bloch’s
theorem, the eigenstates can be chosen to take the form of a plane wave
times a function with the periodicity of the Bravais lattice;
ψk (r + R) = eik·R ψk (r).
(3.4)
where the k is the so called Bloch wave vector. Now, the one-electron
function can be characterized by the Bloch vector k. As a consequence,
Eq. (3.1) can be written as
Heff (r)ψn (k; r) = εn (k)ψn (k; r),
(3.5)
where the index i in Eq. (3.1) has been substituted by the quantum
number n, the band index. The one-electron wave function ψ n and the
9
10
CHAPTER 3. COMPUTATIONAL METHODS
corresponding eigenvalues, εn are now be characterized by the Bloch
wave vector k. The Bloch vector k used to label the one-electron states
is conveniently viewed as a vector in the reciprocal space. A lattice
vector G in the reciprocal space is constructed as
G = x1 b1 + x2 b2 + x3 b3 ,
(3.6)
where the xi are the integers and the bi are the basic vectors of the
reciprocal lattice:
(3.7)
ai · bj = δij .
Regarded as functions of the wave vector k, the energy eigenvalues and
wave functions have the translational symmetry of the reciprocal lattice,
ε(k) = ε(k + G)
(3.8)
ψ(k, r) = ψ(k + G, r).
(3.9)
As the Wigner-Seitz cell is the smallest unit that characterizes the crystal structure, the Brillouin Zone (BZ) is the smallest unit that builds up
the whole reciprocal lattice by repeating itself periodically. By means
of translational symmetry as well as other point group symmetries (rotations, mirror, inversion operations), we can reduce the problem to the
irreducible part of the BZ, the smallest zone which defines a complete
set of wave vectors. For example, in a lattice with full cubic symmetry,
the irreducible part of the BZ is only 1/48 of the full BZ. It is only in
this part we need to solve the electronic structure problem.
According to the Pauli exclusion principle the eigenstates with eigenvalue εi (k) are occupied from the lowest eigenvalue up to the Fermi
energy, εF . The Fermi energy is defined by
N=
εF
−∞
D(ε)dε,
(3.10)
where N is the number of valence electrons and D(ε) is the density of
states (DOS),
dS
2
.
(3.11)
D(ε) = 3
8π S(ε) | ε(k)|
The integration is carried out over a surface of constant energy, S(ε), in
the first BZ. The one electron states most relevant for most of the physical properties are those with energies around the Fermi level. These
states are closely related to crystal structure stability, transport properties, susceptibility, etc..
3.2. ELECTRONIC STRUCTURE METHODS
3.2
11
Electronic structure methods
In practice the solution to equation (3.5), ψ n (k; r), can be expanded in
some basis sets. To solve the problem we need to resort to one of many
available electronic structure methods.
For the different selection of the basis set, electronic structure methods can be divided into two parts [11]:
Fixed basis set
Variable Basis Set
Plane Wave
Augmented Plane Wave
Tight Binding
Korringa Kohn Rostoker
Pseudopotential
Linear Augmented Plane Wave
Orthogonalized Plane Wave
Linearized Muffin Tin Orbitals
Linear Combination of Atomic Orbitals Augmented Spherical Wave
The first set of methods obey the Bloch condition explicitly. That
is, in the expansion
cn φn (r)
(3.12)
ψ(r) =
n
the basis functions are fixed and the coefficients c n are chosen to minimize the energy. One disadvantage of these methods is that the wavefunctions are fixed. This often leads to great difficulty in obtaining a
sufficiently converged basis set.
In the second set of methods, the wavefunctions are varied. This
is performed by introducing energy dependent wavefunctions φ n (ε, r).
The wavefunctions are energy dependent and have the form of
ψ(ε, r) =
cn φn (ε, r),
(3.13)
n
However, the Bloch condition is not automatically fulfilled. The solutions in one unit cell are chosen to fit smoothly to those of the neighbor
cells, thus fulfilling the Bloch condition “indirectly”. As the wavefunction can be modified to the problem at hand, these techniques converge
very fast in the number of required basis function. In APW and KKR,
the price for doing so is the additional parameter ε. At every k-point
of the band structure, equation (3.5) must be solved for a large number
of ε. Solutions only exist for those ε that are actual eigenvalues. While
these methods are accurate, they are also time consuming. The solution
to this problem is to linearize the energy dependent orbitals as is done
in LAPW, LMTO, and ASW. They are expanded as a Taylor expansion
in ε so that the orbitals themselves are energy independent, although
12
CHAPTER 3. COMPUTATIONAL METHODS
Figure 3.1: Muffin-tin part of the crystal potential V(r) and radial wave
function. SMT is the radius of the muffin-tin sphere, S E is the radius of
the escribed sphere and VMTZ is the potential of the interstitial region.
the expansion retains the energy dependence. The variational equation
(3.5) thus has to be solved only once for each k-point. These methods
are extremely rapid and only slightly less accurate than other non-linear
methods.
3.3
The LMTO method
During the last decades, the linear-muffin-tin-orbital (LMTO) [12] method has become very popular for the calculation of the electronic structure of crystalline systems. The LMTO method combines the following
advantages: (1) it uses a minimal basis, which leads to high efficiency
and makes calculations possible for large unit cell; (2) it treats all elements in the same way, so that d and f metals as well as atoms with
a large number of core states can be considered; (3) it is very accurate,
due to the augmentation procedure which gives the wave function the
correct shape near the nuclei; (4) it uses atom-centered basis functions of
well-defined angular momentum, which makes the calculated properties
transparent [13].
3.3.1
Muffin-tin orbitals
The crystal is divided into non-overlapping muffin-tin spheres surrounding the atomic sites and an interstitial region outside the spheres. Inside
the muffin-tin sphere the potential is assumed to be spherically symmetric while in the interstitial region the potential, V MTZ , is assumed to be
3.3. THE LMTO METHOD
13
constant or slowly varying. Because the potential in the interstitial is
constant we can shift the energy scale so as to set it to zero. In the
following, we consider a crystal with only one atom per primitive cell.
Within a single muffin-tin well we define the potential
VMT (r) =
V (r) − VMTZ , |r| < SMT
0
, |r| > SMT
(3.14)
Here V (r) is the spherically symmetric part of the crystal potential. The
radii of the muffin-tin spheres are chosen so that they do not touch each
other. In the following, SMT is expressed by S.
Now we try to solve the Schrödinger equation for muffin-tin potential,
[− 2 +VMT ]ψ(ε, r) = (ε − VMTZ )ψ( r).
We define the kinetic energy
κ2
(3.15)
in the interstitial region by
κ2 = ε − VMTZ
(3.16)
For an electron moving in the potential from an isolated muffin-tin well
embedded in the flat potential VMTZ , the spherical symmetry can extend
throughout all space and the wave functions are
ψL (ε, r) = il Ylm (r̂)ψl (ε, r)
(3.17)
where we use the convention that r = |r| and r̂ is the direction of r. A
phase factor il is included.
To obtain basis functions which are approximately independent of
energy, reasonably localized, and normalizable for all values of κ 2 , Anderson [14] accomplished these by Muffin-tin orbitals. A spherical Bessel
function that cancels the divergent part of ψ l (ε, κ, r) and simultaneously
reduces the energy and potential dependence of the tails, we have the
muffin-tin orbitals in form of
χlm (ε, r) = i
l
Ylm (r̂)
l
(r/S)
ψl (ε, r) + Pl (ε) 2(2l+1)
(r/S)−l−1
, |r| < S
, |r| > S
(3.18)
where ψl (ε, r) is a solution of the radial Schrödinger equation inside the
atomic sphere. The potential function
Dl (ε) + l + 1
(3.19)
Pl (ε) = 2(2l + 1)
Dl (ε) − l
and the normalization of ψl (ε, r) are determined by satisfying differentiability and continuity of the basis function on the sphere boundary.
Here the Dl (ε) is the logarithmic derivative of the wave function. The
tail of the basis function, i.e. the part outside the muffin-tin sphere
can in general be written as Neumann function. But in Eq. (3.16) the
kinetic energy of this tail, known as κ2 , is chosen to be zero. Therefore
the Neumann function has a simple form like this.
CHAPTER 3. COMPUTATIONAL METHODS
14
3.3.2
The LMTO-ASA method
In the atomic sphere approximation, LMTO-ASA, the muffin-tin spheres
are overlapping in such a way that the total volume of muffin-tin sphere
is the same as the atomic volume. This means that the muffin-tin radius
S is equal to the Wigner- Seitz radius S WS where the total volume per
3 . In the ASA, the potential is also
atom is given by V = (4π/3)SWS
assumed to be spherically symmetric inside each muffin-tin sphere and
the kinetic energy of the basis functions defined in the interstitial is
restricted to be constant, actually zero in the calculation.
In order to construct a linear method, the energy dependent terms
in the muffin-tin spheres of the Eq. (3.18) are replaced by the energy
independent function Φ. The function is defined as a combination of
radial functions and their energy derivative
Φ(D, r) = φl (r) + ω(D)φ̇l (r),
(3.20)
where w(D) is a function of the logarithmic derivative and w(D) should
make the energy dependent orbitals χ lm (ε, r) defined in the Eq. (3.18)
continuous and differentiable at the sphere boundary S. The boundary
condition determines D = −l − 1, The so obtained energy independent
orbital can now be written as
χlm (ε, r) = i
3.4
l
Ylm (r̂)
Φl (D, r)
, |r| < S
−l−1
. |r| > S
(r/S)
(3.21)
Full potential LMTO method
The FP-LMTO calculations are all electron, fully relativistic, without
shape approximation to the charge density or potential. The crystal is
divided into non-overlapping muffin-tin sphere and an interstitial region
outside the spheres. The wave function is then represented differently
in the two types of regions. Inside a muffin-tin spheres, the basis functions are as in the LMTO-ASA method. They are Bloch sum of linear
muffin-tin orbitals and are expanded by structure constant, φ ν (r) and
φ̇ν (r). However the kinetic energy is not, as in the ASA approximation,
restricted to the zero in the interstitial region. For simplicity, here we
only consider a monoatomic solid, and suppress the atomic site index.
The κ dependent linear muffin-tin orbitals can now be written as
ψκlm (k, r) = χκlm (r) +
Jκlm (r)Sκlm,l m (k),
(3.22)
lm
where
l
χlm (r) = i
Ylm (r̂)
Φ(Dh ,r)
−iκhl (κS) Φ(D
h ,S)
−iκhl (κr)
, |r| < S
, |r| > S
(3.23)
3.4. FULL POTENTIAL LMTO METHOD
15
and
l
Jκlm (r) = i
Ylm (r̂)
Φ(DJ ,r)
Jl (κS)(κS) Φ(D
J ,S)
Jl (κr)
, |r| < S
. |r| > S
(3.24)
Inside the muffin-tin at τ , we can also expand the electron densities
and potential in spherical harmonics times a radial function,
nτ (r)|τ =
nτ (h; rτ )Dh (r̂τ ),
(3.25)
Vτ (h; rτ )Dh (r̂τ ),
(3.26)
h
Vτ (r)|τ =
h
where Dh are linear combinations of spherical harmonics, Y lm (r̂). Dh
are chosen here because we need an invariant representation of the local
point group of the atomic site contained in the muffin-tin. The expansion
coefficients nτ (h; rτ ) and Vτ (h; rτ ) are numerical functions given on a
radial mesh.
In the interstitial region, the basis function, charge densities and
potential are expressed as Fourier series,
ψ(k; r)|I =
ei(k+G)·r ψ(k + G),
(3.27)
G
nI (r)|I =
nG ei(k+G)·r ,
(3.28)
VG ei(k+G)·r ,
(3.29)
G
VI (r)|I =
G
where G are reciprocal lattice vectors spanning the Fourier space.
3.4.1
The basis set
Envelope function is the basis function in the interstitial region. By
choosing appropriate envelope functions, such as plane waves, Gaussians, and spherical waves (Hankel functions), we can generate various
electronic structure methods (LAPW, LCGO, LMTO, etc.). The LMTO
envelope function is represented as below,
Klm (κ; r) = −κ
l+1 l
i
Ylm (r̂)
2
−h+
l (κr) , κ ≤ 0
, κ2 > 0
nl (κr)
(3.30)
where nl is a spherical Neumann function and h+
l is a spherical Hankel
function of the first kind. The envelope function is a singular Hankel or
Neumann functions with regards to the sign of the kinetic energy. This
CHAPTER 3. COMPUTATIONAL METHODS
16
introduces a κ dependence for the basis functions inside the muffin-tin
sphere through the matching conditions at the sphere boundary. This
is not a problem. Using a variational method, the ground state still has
several basis functions with the same quantum numbers, n, l and m, but
different κ2 . This is called a double basis.
The basis set can always contain different bases corresponding to the
atomic quantum number l but with different principle quantum numbers
n. A basis constructed in this way forms a fully hybridizing basis set,
not a set of separate energy panels.
To illustrate the way the basis set is constructed, we take fcc Ce [15]
as an example. The ground state configuration is 4f 1 5d1 6s2 . Thus we
include the 6s, 6p, 5d, 4f as valence states. To reduce the core leakage
at the sphere boundary, we also treat the core states 5s and 5p as semicore states. By this kind of construction, the basis set becomes more
complete.
3.4.2
The LMTO matrix
We now introduce a convenient notation for the basis functions:
|χi (k) = |φi (k) + |ψi (k),
(3.31)
where |φ is the basis function inside the muffin-tin spheres and |ψ i (k)
represents the basis functions, tails, outside the spheres.
We can construct a wave function Ψ kn (r) by a linear combination of
LMTO basis functions, χi . Hence the linear combination can be written
as
Ai |χi (3.32)
|Ψ =
i
The Hamiltonian operator is
Ĥ = H0 + Vnmt + VI
(3.33)
where H0 is the Hamiltonian operator containing the kinetic operator
and the spherical part of the muffin-tin potential, V nmt represents the
non-spherical part of the muffin-tin potential, and V I is the interstitial
potential. Then by using the variational principle for the one-electron
Hamiltonian the LMTO secular matrix follow as
[χi (k)|H0 + Vnmt + VI |χj (k) − ε(k)χi (k)|χj (k)]Aj = 0
(3.34)
j
We can reduce it to
0
1
[Hij
+ Hij
− ε(k)Oij ]Aj = 0
j
(3.35)
3.4. FULL POTENTIAL LMTO METHOD
17
where
0
= φi (k)|H0 |φj (k)
Hij
(3.36)
Oij = φi (k)|φj (k) + ψi (k)|ψj (k)
(3.37)
1
1
= φi (k)|Vnmt |φj (k) + (κi 2 + κj 2 )ψi (k)|ψj (k) + ψi (k)|VI |ψj (k)
Hij
2
(3.38)
0 is
where |ψj (k) is an eigenfunction to 2 with eigenvalue κ2j . Hij
the spherical muffin-tin part of Hamiltonian matrix. O ij is the overlap
1
between the orbitals inside the sphere as well as in the interstitial. H ij
contains the corrections to the Hamiltonian matrix coming from the
muffin-tin and interstitial region. The first term in Eq. (3.38) is the
non-spherical potential matrix. The next term is the expectation value
of the kinetic energy operator in the interstitial region. The last term is
the interstitial potential matrix.
3.4.3
Total energy
The total energy for whole crystal can be expressed as [16]
Etot = Tval + Tcor + Ec + Exc ,
(3.39)
where Tval and Tcor are the kinetic energy for the valence and core electrons, Ec is electrostatic energy including electron-electron, electronnucleus and nucleus-nucleus energy, and E xc is our familiar term which
has been studied in LDA. The kinetic energy is usually expressed as the
expectation value of the kinetic operator − 2 . By using the eigenvalue
equation the expectation value can be expressed as the sum over one
electron energies minus the effective potential energy. The core eigenvalues εiτ are obtain as exact solution to the Dirac equation with the
spherical part of the muffin-tin potential.
The total energy can be written as
Etot =
occ
kn
−
wnk εkn +
fiτ εiτ +
µτ
1
Zτ j Vc (τj ; 0) +
2 j
Vc
1
n(r)[ Vc (r) − Vin (r)]dr
2
Vc
n(r)εxc (n(r))dr,
(3.40)
where the integral is over the unit cell [14]. The sum j is over the core
states. The density n(r) is the total charge density, valence as well as
core electrons. Vin is the input potential obtained from LDA. Madelung
term Vc (τ ; 0) is the Coulomb potential at the nucleus less the Z/r self
contribution and εxc is the exchange-correlation energy.
18
3.5
CHAPTER 3. COMPUTATIONAL METHODS
Projector Augmented Wave Method
Blöchl [17] developed the projector augumented wave method (PAW)
by combining the ideas from pseudopotentials and linear augmentedplane-wave (LAPW) methods. PAW method is an all-electron electronic
structure method. It describes the wave function by a superposition of
different terms: the plane wave part, the so-called pseudo wave function,
and expansions into atomic and pseudo atomic orbitals at each atom.
On one hand, the plane wave part has the flexibility to describe the
bonding and tail region of the wave functions, but if it is used alone it
would require prohibitive large basis sets to describe correctly all the oscillations of the wave function near the nuclei. On the other hand, the expansions into atomic orbitals can describe correctly the nodal structure
of the wave function near the nucleus, but lack the variational degrees of
freedom for the bonding and tail regions. The PAW method combines
the virtues of both numerical representations in one well-defined basis
set.
To avoid the dual efforts by performing two electronic structure calculations, both plane waves and atomic orbitals, the PAW method does
not determine the coefficients of the atomic orbitals variationally. Instead, they are unique functions of the plane wave coefficients. The total
energy, and most other observable quantities can be broken into three
almost independent contributions: one from the plane wave part and a
pair of expansions into atomic orbitals on each atom. The contributions
from the atomic orbitals can be broken down furthermore into contributions from each atom, so that strictly no overlap between atomic orbitals
on different sites need to be computed.
In principle, the PAW method is able to recover rigorously the density functional total energy, if plane wave and atomic orbital expansions
are complete. This provides us with a systematic way to improve the
basis set errors. The present implementation uses the frozen core approximation, it provides the correct densities and wave functions, and
thus allows us to calculate other parameters of the system. By making
the unit cell sufficiently large and decoupling the long-range interactions,
limitations of plane wave basis sets to periodic systems (crystals) can
easily be overcome. Thus this method can be used to study molecules,
surfaces, and solids within the same approach.
3.5.1
Wave function
Firstly, we will introduce a transformation matrix τ . There are two
Hilbert spaces, one called all electron (AE) Hilbert, and the other called
3.5. PROJECTOR AUGMENTED WAVE METHOD
19
pseudo (PS) Hilbert. We need to map the AE valence wave functions
onto to the fictitious PS wave functions.
Every PS wave function can be expanded into PS partial waves
|Ψ̃ =
|φ̃i ci
(3.41)
i
The corresponding AE wave function is of the form
|Ψ = τ |Ψ̃ =
|φi ci
(3.42)
i
From the two equations above, we can derive
|Ψ = |Ψ̃ −
|φ̃i ci +
i
|φi ci
(3.43)
i
because we need the transformation τ to be linear, the coefficients must
be linear functions of the PS wave functions. Therefore the coefficients
are scalar products of PS wave function with projector functions p̃ i |,
p̃i |Ψ̃. The projector functions must fulfill the condition
|φ̃i p̃i | = 1
(3.44)
i
within the augmentation region ΩR , which implies that
p̃i |φ̃j = δij .
(3.45)
Finally, the transformation matrix can be deduced from Eq. (3.42) and
Eq. (3.43) with the definition ci = p̃i |Ψ̃
τ =1+(
|φi − |φ̃i )p̃i |.
(3.46)
i
Using this transformation matrix, the AE valence wave function can be
obtained from PS wave function by
|Ψ = |Ψ̃ +
(|φi − |φ̃i )p̃i |Ψ̃
(3.47)
i
The core states wave functions |Ψc are decomposed in a way similar
to the valence wave functions. They are decomposed into three contributions:
(3.48)
|Ψc = |Ψ̃c + |φc − |φ̃c .
Here |Ψ̃c is a PS core wave function, |φc is AE core partial wave and
lastly |φ̃c is the PS core partial wave. Comparing to the valence wave
functions no projector functions are needed to be defined for the core
states, and the coefficients of the one-center expansion are always unity.
CHAPTER 3. COMPUTATIONAL METHODS
20
Figure 3.2: PAW method illustration
3.5.2
Charge density
The charge density at point r in space is composed of three terms:
n(r) = ñ(r) + n1 (r) − ñ1 (r)
(3.49)
The soft pseudo charge density ñ(r) is the expectation value of real-space
projection operator |rr| on the pseudo-wave-functions.
ñ(r) =
fn Ψ̃n |rr|Ψ̃n (3.50)
n
The onsite charge densities n1 and ñ1 are treated on a radial support
grid. They are given as:
n1 (r) =
fn Ψ̃n |p̃i φi |rr|φj p̃j |Ψ̃n = ρij φi |rr|φj (3.51)
n
here ρij is the occupancies of each augmentation channel (i, j) and they
are calculated from the pseudo-wave-functions applying the projector
functions: ρij = n fn Ψ̃n |p̃i p̃j | Ψ̃n , and
ñ1 (r) =
fn Ψ̃n |p̃i φ̃i |rr|φ̃j p̃j |Ψ̃n = ρij φ̃i |rr|φ̃j (3.52)
n
We will focus on the frozen core case, ñ, ñ 1 , n1 are restricted to the
valence quantities. Besides that, we introduce four quantities that will be
used to describe the core charge density: n c , ñc , nZc , ñZc . nc denote the
charge density of frozen core all-electron wave function in the reference
atom. The partial core density ñ is introduced to calculate nonlinear
core corrections. nZc is defined as the sum of the point charge of nuclei
nZ and frozen core AE charge density n c : nZc = nZ + nc ,
Lastly, the pseudized core density is a charge distribution that is
equivalent to nZc outside the core radius and have the same moment as
the nZc inside the core region.
nZc (r)d3 r =
Ωr
ñZc (r)d3 r
Ωr
(3.53)
3.5. PROJECTOR AUGMENTED WAVE METHOD
21
The total charge density nT [18] is decomposed into three terms:
= n + nZc
nT
= (ñ + n̂ + ñZc ) + (n1 + nZc ) − (ñ1 + n̂ + ñZc )
= ñT + n1T − ñ1T
(3.54)
A compensation charge n̂ is added to the soft charge densities ñ+ ñ Zc
and ñ1 + ñZc to reproduce the correct multipole moments of the AE
charge density n1 + nZc that is located in each augmentation region.
Because nZc and ñZc have exactly the same monopole −Zion (charge
of an electron is +1), the compensation charge must be chosen so that
ñ1 + n̂ has the same moments as the AE valence charge density n 1 within
each augmentation sphere.
3.5.3
Total energy
The final expression for the total energy can also be split into three
terms:
(3.55)
E(r) = Ẽ(r) + E 1 (r) − Ẽ 1 (r).
Where Ẽ(r), E 1 (r), Ẽ 1 (r) are given by
Ẽ(r) =
n
1
fn Ψ̃n | − ∆|Ψ̃n + Exc [ñ + n̂ + ñc ] + EH [ñ + n̂]
2
vH [ñZc ][ñ(r) + n̂(r)]dr + U (R, Zion )
+
(3.56)
U (R, Zion ) is the electrostatic energy of point charges Z ion in an uniform
electrostatic background,
1
ρij φi | − ∆|φj + Exc [n1 + nc ] + EH [n1 ]
E 1 (r) =
2
i,j
vH [ñZc ]n1 (r)dr
+
(3.57)
Here vH [ñZc ]n1 (r)dr is the electrostatic interaction between core and
valence electrons and EH is electrostatic energy
1
1
EH [n] = (n)(n) =
2
2
Ẽ 1 (r) =
i,j
+
dr
dr
n(r)n(r )
|r − r |
(3.58)
1
ρij φ̃i | − ∆|φ̃j + Exc [ñ1 + n̂ + ñc ] + EH [ñ1 + n̂]
2
vH [ñZc ][ñ1 (r) + n̂(r)]dr
(3.59)
The overline means that the corresponding terms must be evaluated on
the radial grid within each augmentation region.
CHAPTER 3. COMPUTATIONAL METHODS
22
3.6
Ultrasoft pseudopotential
It is unaffordable to treat first-row elements, transition metals, and rareearth elements by standard Norm-conserving Pseudopotentials (NCPP). Therefore, various attempts have been made to generate the so
called soft potentials, and Vanderbilt [19] ultrasoft pseudopotentials
(US-PP) has been proved to be the most successful one among them.
There are number of improvements in US-PP method: 1) nonlinear core
corrections were included in the US-PP. 2) Lower cutoff energy, namely
reduced number of plane waves, was required in US-PP than NC-PP.
This enables us to perform molecular dynamics simulations for systems
containing first-row elements and transition metals.
is exactly the same in the PAW method and US-PP meBecause E
thod, we only need to consider the linearization of E 1 and Ẽ 1 . We obtain
E 1 to the first order by linearization of the E 1 in the PAW total energy
functional around atomic reference occupancies ρ ij
E1 ≈ C +
ij
1
a
ρij φi | − ∆ + υef
f |φj 2
(3.60)
a
1
1
with υef
f = υH [na + nZc ] + υxc [na + nc ] and C is a constant.
A similar linearization can also be done for Ẽ 1
Ẽ ≈ C +
1
1
a
[ρij φi | − ∆ + υef
f |φj +
2
ij
a
L (r)υ
ef
Q
ij
f (r)dr]
(3.61)
with
a
1a + n
a + n
Zc] + υxc [n
1a + n
a + n
Zc ]
υef
f = υH [n
(3.62)
L (r) is a pseudized augmentation charge in the US-PP approaches.
Q
ij
∗ (r)φ
j (r),
L (r) = Qij (r) = φ∗ (r)φj (r) − φ
Given Q
ij
i
i
1 =
E1 − E
ij
1
1
ρij (φi | − ∆|φj − φi | − ∆|φj ).
2
2
(3.63)
Now, we can compare the PAW functional with the US-PP functional. In the PAW method, if the sum of compensation charge and
1 + n
, is equivalent to the onsite AE charge denpseudocharge density, n
1
1 from
Zc = nZc , n
c = nc ,we can derive the same E1 − E
sity n , and n
Eq. (3.57) and Eq. (3.59). In this limiting case, The PAW method is
equivalent to the US-PP method.
3.7. PAW AND US-PP
3.7
23
PAW and US-PP
The general rule in Vienna ab initio simulation package (VASP) is to
use PAW potential wherever possible, the PAW potentials are especially
generated for improving the accuracy for magnetic materials, alkai and
alkai earth elements, 3d transition metals, lanthanides and actinides.
For these materials, the treatment of semicores states as valence states
are desirable. The PAW method is as efficient as the FLAPW method,
it is easy to unfreeze of low lying core states, only one partialwave (and
project) for the semicore states is included.
Differences between PAW and US-PP are only related to the pseudization of the augmentation charges. By choosing very accurate pseudized
augmentation function, discrepancies of both methods can be removed.
However, augmentation charges must be represented on a regular grid
with the US-PP approach. Therefore, hard and accurate pseudized augmentation charges are expensive in terms of computer time and memory.
The PAW method avoids these drawbacks by introducing radial support
grids. The rapidly varying functions can be elegantly and efficiently
treated on radial support grids.
The PAW potentials are generally slightly harder than US-PP and
they retains similar hardness across the periodic table. Vice versa, the
US-PP Potential become progressively softer when moving down in the
periodic table. For multi species compounds with very different covalent
radii mixed, the PAW potentials are clearly superior, except for one
component system, the US -PP might be slightly faster at the price of
reduced precision. Most PAW potential were optimised to work at a
cutoff of 250-300 eV, which is only slightly higher than in the US-PP.
24
CHAPTER 3. COMPUTATIONAL METHODS
Chapter 4
Phase transitions
4.1
Static total energy calculation
Static total energy is still the main method of ab initio simulation. These
calculations evaluate the energy and stability of an ‘ideal’, zero temperature crystal in which all atoms are located on their lattice positions.
Pressure induced phase transitions can be reliably predicted by evaluating the enthalpy (total energy plus PV) for each phase as a function of
pressure.
At a given pressure, the stable structure is the one which has the
lowest minimum enthalpy. The phase transition pressure can also be
deduced from the common tangent between curves on a total energy vs
volume graph corresponding to the two phases. The transition pressure
is given by PT = (F2 − F1 )/(V1 − V2 ) where F1 and F2 are the Helmholtz
free energy for phase 1 and 2 respectively (this is identical to the total
energy for T = 0). The free energy is minimised with respect to the
internal coordinates and unit cell parameters in each phase. One cannot
evaluate PT directly from the above equation. So one has to calculate
equations of states for the two phases separately, and then compared.
The hydrous phases δ-AlOOH (paper XI) have recently been subjected to various studies at high pressure and high temperature [20, 21].
One of the most important hydrous phase is the so called Egg phase,
AlSiO3 OH [22, 23]. Recent x-ray diffraction studies at high pressure
and high temperature have shown that this Egg phase decomposes into
δ-AlOOH and stishovite SiO2 at 23 GPa and 1000◦ C. This decomposition reaction suggests that water stored in the phase Egg can be carried
further by δ-AlOOH into the deep lower mantle. The stability field of
δ-AlOOH was reported to be from 17 GPa up to at least 25 GPa at
around 1000◦ C to 1200◦ C [24]. In our calculation, a phase transition
from α-AlOOH to δ-AlOOH was calculated to take place at 17.9 GPa
25
CHAPTER 4. PHASE TRANSITIONS
26
Difference in enthalpy (mRy/atom)
3
δ-AlOOH
α-AlOOH
2
1
0
-1
-2
-3
0
10
20
30
40
50
Pressure (GPa)
Figure 4.1: The 0 K enthalpy as a function of pressure for two different
crystal structures: α-AlOOH and δ-AlOOH. The enthalpy of the αAlOOH phase is taken as the energy zero. The transition from α-AlOOH
to δ-AlOOH takes place at 17.9 GPa.
and with a volume collapse of 3%, as shown in Fig. 4.1.
4.2
Elastic stability criteria
For a cubic crystal with elastic constants C 11 , C12 and C44 , the generalized elastic stability criteria are C 11 + 2C12 > 0, C44 > 0 and
C11 − C12 > 0 [25, 26]. The transition metals, Ti, Zr and Hf stabilized in the hcp structure at the ambient conditions. The bcc phase is
calculated to be unstable and with a negative C (C = (C11 − C12 )/2).
With the increasing pressure, C becomes positive at the V /V0 = 0.73. A
high pressure bcc phase is predicted to be stable, this can be understood
as an effect of the s → d transfer under compression. This means that
Ti behave more like its nearest neighbor V, which has the bcc crystal
structure [27] at ambient conditions.
4.3
Bain path
The structural path for going from bcc phase to fcc phase within the
tetragonal structure is well known as Bain Path [28]. Body-centered
and face-centered cubic crystal can be considered as special
cases of a
√
body-centered tetragonal crystal with c/a = 1 and 2, respectively.
Starting with a normally fcc element Ce [29] (Fig. 4.2), and calculating
energy as a function of c/a, we obtain two local minimums at c/a = 1.41
4.4. DYNAMICAL STABILITY (PHONON CALCULATION)
27
V/V0=0.57
Total Energy (Ry)
-0.7025
V/V0=0.58
-0.706
1.4
1.5
1.6
c/a ratio
Figure 4.2: The comparison between the c/a ratio of V /V 0√= 0.57 and
V /V0 = 0.58 for Ce. When V /V0 = 0.58, the fcc (c/a = 2) phase is
more stable. When V /V0 = 0.57, the bct phase is more stable.
and c/a = 1.66. The curvature of E(c/a) around the maximum and the
minimum directly correspond to√ C we mentioned above. Before the
transition, the fcc phase (c/a = 2) is more stable compared to the bct
phase and after the transition, the bct phase starts to win in the total
energy as the volume is changing from V /V 0 = 0.58 to V /V0 = 0.57.
4.4
Dynamical stability (phonon calculation)
The soft phonon phase transition is one of the best established mechanisms by which a crystal structure can change [30]. In the pressureinduced case, the frequency of a given vibration in the lattice goes to zero
as the transition is approached: zero frequency implies that the lattice
structure has become unstable, and will transform to a new phase.
Considering a system consists of N atoms, the Hamiltonian of the
system can then be expressed as [31]
H=
1 1
Mi [u̇(i)]2 +
φαβ (i, j)uα (i)uβ (j)
2 i
2 ij αβ
(4.1)
Here mi is the mass of atom i and u(i) is its displacement away from
its equilibrium position, while α and β subscripts denote one of the
Cartesian components of a vector. φαβ (i, j) is the so called force tensor,
which is simply the second derivative of the potential.
φαβ (i, j) =
∂2U
∂uα (i)∂uβ (j)
(4.2)
CHAPTER 4. PHASE TRANSITIONS
28
The substitution e(i) =
H=
√
Mi u(i) yields
1
φ(ij)
ė(i)2 +
e(i) e(j)
2 i
Mi Mj
ij
(4.3)
We define the dynamical matrix for the system,
1
Dαβ (i, j) = φαβ (i, j)
Mi Mj
(4.4)
which can be constructed from force constant tensor. The size of the
dynamical matrix is 3N × 3N . Diagonalizing the dynamical matrix we
obtain all of its eigenvalues λm , m = 1 · · · 3N . In the harmonic approximation, the knowledge of these frequencies is sufficient to determine
other thermodynamic quantities of the system. For example, the free
energy of the system can now be calculated through [32]
F =
3N
kB T h̄ωm
)].
ln[2sinh(
N m=1
2KB T
(4.5)
In a crystal, the determination of the normal modes is somewhat
simplified by the translational symmetry
of the system. Say n denotes
l
the number of atoms per unit cell, u
is the displacements of atom
i
l l
is the force
i in cell l away from its equilibrium position, and Φ
i j
constant relative to atom i in cell l and atom j in cell l and let
e
l
i
= exp [ι2π(k · l)] e
0
i
,
(4.6)
√
where ι = −1, l denotes the Cartesian coordinates of one corner of
cell l, and k is a point in the first Brillouin zone. This fact reduces
the problem of diagonalizing the 3N × 3N matrix D to the problem of
diagonalizing a 3n × 3n matrix D(k) for various values of k. This can
be shown by a simple substitution of Eq. (4.6) into Eq. (4.3). The
dynamical matrix D(k) to be diagonalized is given by






exp[ι2π(k · l)] 
D(k) =


l



Φ
√
Φ
√
o l
1 1
o l
n 1
M1 Mn
···
..
.
···
o l
1 n





M1 Mn

..

.



o l

Φ

n n
Φ
M1 Mn
..
.
√
√
M1 Mn
(4.7)
4.5. EQUATION OF STATE
29
As before, the resulting eigenvalues λ(k) for i = 1, · · ·, n give the fre1
λ(k). The function ω(k) for
quencies of the normal modes ω(k) = 2π
a given i is called a phonon branch, while the plot of the k dependence
of all branches along a given direction in k space is called the phonon
dispersion curve. In periodic systems, the phonon DOS, which gives the
number of modes of oscillation having a frequency lying in the energy
interval [ω, ω + dω] is defined as
g(ω) =
3n i=1 BZ
δ[ω − ωi (k)]dk
(4.8)
where the integral is taken over the first Brillouin zone. Phonon calculation in Ti was performed by means of PWSCF package (Plane-Wave
Self-Consistent Field). PWSCF is a first-principles energy code that uses
pseudopotentials (PP) and ultrasoft pseudopoentials (US-PP) within
DFT. In contrast to the frozen-phonon method, it includes linear response method, which allows the treatment of arbitrary phonon wave
vectors q. The phonon dispersion curves along several of high symmetry
k points for the Ti high pressure γ phase is presented. Vohra et al. [33]
observed a transformation from an ω phase to an orthorhombic phase
(γ phase) at a pressure of 116 ± 4 GPa, and this phase is stable up to
146 GPa. In this pressure range (see Fig. 4.3), our calculated phonon
dispersion curves give an imaginary phonon frequency indicating that
the γ phase is unstable in the pressure range we calculated, from 119
GPa to 173 GPa.
4.5
Equation of state
The equation of state (EOS) of solids is of great importance in basic
and applied science. The measurable properties of solids, such as the
equilibrium volume (V0 ), the bulk modulus (B0 ) and its first pressure
derivative (B0 ) are directly related to the EOS. The high pressure EOS
has been represented in various functional forms, for example, the Murnaghan equation, Birch-Murnaghan (BM) equation, universal equation
and recently a new equation of state which is appropriate at strong
compressions has been put forward by Holzapfel [34, 35].
4.5.1
Murnaghan equation of state
Murnaghan EOS was introduced by Murnaghan et al. in the forties [36].
The E-V form Murnaghan EOS can be represented as,
E(V ) =
B0 V0
B0
1
B0 −1
V0
V
B −1
0
+
V
V0
−
B0
B0 −1
+ Ecoh
(4.9)
CHAPTER 4. PHASE TRANSITIONS
30
16
12
8
4
Frequency (THz)
0
16
12
8
4
0
16
12
8
4
0
Γ
Z
T
Γ
Y
S
R
Figure 4.3: The phonon dispersion curves for the γ phase of Ti: from
up to down, P=119, 140 and 173 GPa respectively.
Ecoh is the cohesive energy and is treated as an adjustable parameter.
Since the pressure can be obtained from P (V ) = −∂E(V )/∂V , the Murnaghan equation can be expressed in its usual form
P (V ) =
B0
B0
V0
V
B 0
−1 .
(4.10)
The bulk modulus is derived through the volume derivative of the equation above, B = −V (∂P /∂V ),
B(V ) = B0
4.5.2
V0
V
B 0
.
(4.11)
Birch-Murnaghan equation of state
Birch et al. [37, 38] expanded the Gibb’s free energy F in terms of
Eulerian strain , with V0 /V = (1 − 2)3/2 . The integrated energyvolume form of the third order BM-EOS,
V3
V
7/3
9
B0 [(4 − B0 ) V02 − (14 − 3B0 ) V04/3
E(V ) = − 16
V
5/3
+(16 − 3B0 ) V02/3 ] + E0
(4.12)
Using the obtained B0 , B0 and V0 from a least-square fit of the calculated
V-E curves to the EOS above, the hydrostatic pressure P was determined
4.5. EQUATION OF STATE
31
from the P-V form of the BM EOS, which is the volume derivative of
the former equation. The second order BM-EOS can be written as
P (V ) = 1.5B0
V0
V
7/3
−
V0
V
5/3 (4.13)
While the third order BM-EOS [39] has the analytical form;
P (V ) = 1.5B0
· 1+
3
4 (B0
V0
V
− 4)
7/3
−
2/3
V0
V
V0
V
5/3 −1
(4.14)
The bulk modulus corresponding to Eq. (4.14) is
B(V ) = 1.5B0
7
3
V0
V
· 1 + 34 (B0 − 4)
+1.5B0
4.5.3
V0
V
7/3
−
V0
V
7/3
V0
V
−
2/3
5/3 5
3
V0
V
5/3 −1
1
2 (B0
− 4)
V0
V
2/3 (4.15)
Universal equation of state
Vinet et al. [40] have reported a universal form the EOS for all classes
of solids, such as ionic, metallic, covalent and rare-gas solid, under compression. Their P-V relation can be represented as [41]
(1 − x)
exp[η(1 − x)]
P (V ) = 3B0
x2
(4.16)
Here η is fixed in terms of B0
η = 3/2(B0 − 1)
(4.17)
and x = ( VV0 )1/3 . Poirier [42, 43] has pointed out that the Universal
EOS can be obtained by the same derivation as the Birch-Murnaghan
EOS, using a strain parameter = (V0 /V )1/3 − 1 and the free energy
F = F0 (1 + A)exp(−A) (F0 and A are constants).
4.5.4
Comparison of different EOS for Fe
Iron has attracted a lot of attention from geophysicists because it is
considered to be the major constituent of the earth core. At low temperature, a Fe bcc (α) - hcp () transition takes place around 13 GPa
[44], and the hcp phase is believed to be stable up to the pressure of
CHAPTER 4. PHASE TRANSITIONS
32
the inner core. In paper IX, we show that the c/a ratio of Fe increases
with increasing pressure and eventually approaches the ideal ratio of
8
3 = 1.633 at extreme pressures. Three different equations of states
(as shown in Fig. 4.4) have been used to fit the same set of GGA calculated E-V data set for the hcp phase. Because B 0 and B0 are highly
correlated parameters in the Birch-Murnaghan equation of state (EOS)
[45], we show the comparison when B0 is taken to be the experimental value 5.8 [46]. These three EOS have similar behavior in the low
pressure range, but they start to diverge at a pressure around 150 GPa.
The Vinet EOS shows a better agreement with the experimental data
of Mao et al. and Dubrovinsky et al. under pressure. All these EOS
fitted equilibrium volume (6.14 cm3 /mol) is lower than that obtained
from the experimental data (6.73 cm 3 /mol [46]). The treatment of antiferromagnetically ordered structure gives a better equilibrium V 0 =6.35
cm3 /mol [47]. However, none of them can successfully reproduce the
experimental equilibrium volume.
500
450
Jephcoat et al.
Mao et al.
Dubrovinsky et al.
Vinet EOS
Murnaghan EOS
BM EOS
Fe
400
Pressure (GPa)
350
300
250
200
150
100
50
0
4
4.5
5
5.5
6
6.5
3
Volume (cm /mol)
Figure 4.4: Comparison between the experimental and theoretical EOS
using three different EOS for hcp Fe. The filled circle data points are
experiments from Mao et al. [48] up to 304 GPa without medium. The
square data points correspond to 300 K x-ray diffraction measurements
with Ar and Ne medium up to to 78 GPa [49]. The dot-dashed line is
the latest data from Dubrovinsky et al. [50]. The Murnaghan, Birchmurnaghan and Vinet EOS are denoted by dashed, dotted and solid line
respectively.
Chapter 5
Semiconductor optics
5.1
Dielectric function
Materials having an energy band gap, Eg , in the range 0 < Eg ≤ 4
eV are called semiconductors and those where having a gap E g ≥ 4
eV are insulators [51]. Semiconductors with a gap approximately below
or near 0.5 eV are named as narrow-gap semiconductors; on the other
hand, materials with a gap between 2 eV and 4 eV are called wide-gap
semiconductors. If Eg is close to zero, they are called semimetals, such
as TiC.
It is known that the measurement of optical properties is helpful to explain the electronic structure of materials. The knowledge
of the refractive indice and absorption coefficient of semiconductors
is especially important in the design and analysis of heterostructure
lasers and other semiconductor devices [52]. The dielectric function,
ε(w) = ε1 (w) + iε2 (w), fully describes the optical properties of medium
at all photon energies, h̄ω. The (q = 0) dielectric function can be calculated in the momentum representation, which requires matrix elements of the momentum, p, between occupied and unoccupied eigenstates. To be specific, the imaginary part of the dielectric function,
ε2 (w) = Imε(q = 0, w), can be calculated from
εij
2 (w) =
4π 2 e2
Ωm2 w 2
knn σ
< knσ|pi |kn σ >< kn σ|pj |knσ >
·fkn (1 − fkn )δ(ekn − ekn − h̄w).
(5.1)
In Eq. (5.1), e is the electron charge, m is its mass, Ω is the crystal volume, and fkn is the Fermi distribution. Moreover |knσ is the
crystal wave function corresponding to the nth eigenvalue with crystal
momentum k and spin σ. With our spherical wave basis functions, the
matrix elements of the momentum operator are conveniently calculated
in spherical coordinates and for this reason the momentum is written in
33
CHAPTER 5. SEMICONDUCTOR OPTICS
34
√
p = µ e∗µ pµ , where µ is -1, 0, or 1, and p−1 = 1/ 2(px − ipy ), p0 = pz ,
√
and p1 = 1/ 2(px + ipy ).
The evaluation of the matrix elements in Eq. (5.1) is done over the
muffin-tin region and the interstitial separately. The integration over the
primitive cell is done in way similar to what Oppeneer [53] and Gasche
[11] did in their calculations.
The summation over the Brillouin zone in Eq. (5.1) is calculated using linear interpolation on a mesh of uniformly distributed points, i.e.,
the tetrahedron method. Matrix element, eigenvalues, and eigenvectors
are calculated in the irreducible part of the Brillouin-zone. The correct symmetry for the dielectric constant is obtained by averaging the
calculated dielectric function.
Through the Kramers-Kronig relations, we have derived the real part
of the dielectric function ε1 .
ε1 (w) = 1 +
1
π
∞
0
dω ε2 (ω )(
ω
1
1
+ )
−ω ω +ω
(5.2)
The relation between the dielectric function and the complex refractive index N = n + ik is given by
ε1 = n2 − k2
(5.3)
ε2 = 2nk
(5.4)
and
Given n and k, we can derive the normal-incidence reflectivity R
R=
(n − 1)2 + k2
(n + 1)2 + k 2
(5.5)
and the absorption coefficient α
α=
5.2
4πk
λ
(5.6)
Dielectric function of PbWO4
Under the ambient conditions, lead tungstate has two stable crystallographic structures, raspite and scheelite. For raspite, with a monoclinic
structure, the total dielectric function is composed of three different dielectric functions along the a, b and and c axis. But for scheelite, having
a tetragonal structure, we only need to average over two components to
get the total dielectric function, namely the components corresponding
5.2. DIELECTRIC FUNCTION OF PBWO4
35
to light polarized parallel and perpendicular to the c axis. In this case
the total, orientation averaged ε2 is given by
εtot
2 (w)
ε (w) + 2ε⊥
2 (w)
= 2
3
(5.7)
⊥
In the discussion that follows we will refer to ε ij
2 (w) as ε2 (w) when
i=j=x and as ε2 (w) when i=j=z.
20
6
5
4
3
15
10
(a) scheelite PbWO4
A↑
↑
B
3.3
3.4
3.5
ε2
5
0
(b) raspite PbWO4
3
2
15
10
0
experiment
calculation
C↓
1
2.6
2.8
3
5
0
0
5
10
15
20
25
PHOTON ENERGY (eV)
Figure 5.1: The imaginary part of the dielectric function for both scheelite (a) and raspite structure (b) of PbWO 4 . The experimental data are
from Itoh et al. [54]. The inserts show an expanded part of ε 2 .
In Fig. 5.1, we show the averaged ε2 for both scheelite and raspite
structure. Our calculated dielectric function agrees very well with the
experimental data even without any broadening. From the experimental
dielectric function, we can see the energy gap, around 3.5 eV acts as a
threshold for interband excitations for both structures. In the scheelite
structure, the doublet peaks observed by Shpinkov et al. [55] and Itoh et
al. can be clearly seen at 3.30 eV and 3.44 eV from our calculation. The
sharp intensive experimental peak at 4.18 eV which these two doublet
peaks contribute to was reproduced by a rather prominent calculated
peak at 3.35 eV. According to the selection rule, l = ±1, only such
transitions that change of the angular momentum quantum number of
1 are allowed. These two peaks in the calculated spectrum can be assigned to Pb s → p interband transitions. In the raspite structure, most
features of the measured data are well reproduced by our calculations.
36
CHAPTER 5. SEMICONDUCTOR OPTICS
The weak peak at 3.75 eV was reproduced by a calculated peak at 2.95
eV in Fig. 5.1 (b). Since the energy gap was underestimated by around
1 eV in the LDA approximation, and if we shift the calculated curve by
1 eV, the agreement will be even better.
5.3
Solar energy materials
Solar cells are devices which convert solar energy into electricity, either
directly via the photovoltaic effect, or via the intermediate of heat or
chemical energy. The ideal solar energy cell is required to have a band
gap in the visible light region as well as to have high absorption in the
visible region. CuInSe2 (CIS) has a band gap of only 1 eV, with Ga
addition to CIS, forming the CuIn1−x Gax Se2 (CIGS) alloy, the gap is
raised, and thus increases the open circuit voltage [56]. At present, the
putatively best CIGS solar cells are made with 30% doping of Ga [57].
The effect of the Ga doping to CuInSe2 can be observed in the change
of the band gap [58]
Eg (x) = (1 − x)Eg (CIS) + xEg (CGS) − bx(1 − x),
(5.8)
b=0.21 was given by Wei et al. [56] CIS exhibits a direct band gap at
Γ point, which indicates that the photon energy is directly converted
into the creation of electron/hole pairs in the semiconductor material.
This is preferable to the material that exhibits an indirect band gap,
where there is some energy loss due to the creation of phonons. The
band gap in CIS is 1.04 eV [59], and 1.68 eV [59] in CGS, according
to Eq. (5.8) CuIn0.75 Ga0.25 Se2 , CuIn0.5 Ga0.5 Se2 should have the band
gap around 1.16 eV and 1.31 eV respectively. Although the chalcopyrite structure resembles to the zinc-blende structure, the band gaps of
the ternary (I-III-VI2 ) semiconductors are only less than half of their
binary analogs (II-VI). This band gap anomaly makes CIS one of the
most known absorbers in the solar spectrum. The p-d hybridization and
structure anomaly attribute to the anomaly in the band gap [59]. From
upper panel of Fig. 5.2, we can easily observe the opening of the band
gap with Ga addition to CuInSe2 . The band gap is directly obtained
from calculation without any shift.
CIS has a high optical absorption coefficient, which provides information about optimum solar energy conversion efficiency. The absorption
coefficient of a material indicates how far light of a specific wavelength
(or energy) can penetrate into the material before being absorbed.
5.3. SOLAR ENERGY MATERIALS
12
37
E⊥c
10
CuInSe2
CuIn0.75Ga0.25Se2
CuIn0.5Ga0.5Se2
8
CuGaSe2
6
4
ε2
2
0
12
E || c
10
8
6
4
2
0
2
4
8
6
PHOTON ENERGY (eV)
Figure 5.2: The imaginary part of the dielectric function of CuInSe 2 ,
CuIn0.75 Ga0.25 Se2 , CuIn0.5 Ga0.5 Se2 and CuGaSe2 for both E⊥c and Ec.
8
5
Absorption coeifficient (×10 )
E⊥ c
CuInSe2
6
4
2
0
8
E || c
6
4
2
0
0
1
2
3
Energy (eV)
4
5
Figure 5.3: The absorption coefficient of CuInSe 2. The experimental
data and theoretical results are denoted by dashed and solid line respectively. The experimental data are taken from Kawashima et al. [60].
38
CHAPTER 5. SEMICONDUCTOR OPTICS
A small absorption coefficient means that light is not readily absorbed by
the material. The depth of penetration (1/α = d in this case) is defined
by the distance at which the radiant power decreases to 1/e of its incident
value. The fundamental absorption corresponds to a strong absorption
region which is in order of 105 cm−1 to 106 cm−1 . The fundamental
absorption area is manifested by the rapid rise in absorption and is used
to determine the energy gap of the semiconductor. In Fig. 5.3, the
interband transitions start at around 1 eV, and CuInSe 2 shows a high
absorption coefficient up to 8 × 105 cm−1 in the energy range where we
are interested in.
Chapter 6
MAX phases
6.1
MN+1 AXN phase
MN+1 AXN (MAX) (N=1 to 3) phases are a series of ceramics but with a
combination of ductility, conductivity and machinability comparable to
metals. M is an early transition metal, A is an A-group element (mostly
IIIA and IVA) and X is either C or N. These phases have hexagonal
layered structures and belong to the space group P6 3 /mmc, wherein
MN+1 XN layers have the rock salt structure interleaved with pure layers
of the A-group elements (Fig. 6.1). The structures of the vast majority
of these compounds were determined by Nowotny [61] and co-workers
in the sixties. Up tp now, there are roughly fifty M 2 AX phases, three
M3 AX2 and one M4 AX3 phases known [7] (Table 6.1).
Figure 6.1: Crystal structures of 211, 312 and 413.
39
CHAPTER 6. MAX PHASES
40
Table 6.1: Known MAX phases
211 Ti2 AlC
Nb2 AlC
Nb2 SnC
Zr2 PbC
Hf2 SC
Nb2 SC
Sc2 InC
Ti2 CdC
Ti2 AlN (Nb,Ti)2 AlC
Ti2 PbC
Nb2 PC
Ti2 GaN
Cr2 GaN
Hf2 SnN
Hf2 InN
Ti3 AlC2
312 Ti3 SiC2
413 Ti4 AlN3
6.2
Ti2 GeC
Zr2 SnC
Hf2 PbC Ti2 AIN0.5 C0.5
Ti2 GaC
V2 GaC
Ti2 InC
Zr2 InC
Cr2 AlC
Ta2 AlC
Cr2 GaC
Nb2 GaC
V2 GaN
V2 GeC
Ti2 TlC
Zr2 TlC
Ti3 GeC2
Hf2 SnC
Zr2 SC
V2 AsC
Nb2 InC
V2 AlC
Mo2 GaC
Ti2 InN
Hf2 TlC
Ti2 SnC
Ti2 SC
Nb2 AsC
Hf2 InC
V2 PC
Ta2 GaC
Zr2 InN
Zr2 TlC
Phase stability in Ti-Si-C system
In the paper XII, we presented the formation energy calculation for
Ti-Si-C system. The phase stability has been predicted by comparing
the cohesive energy of the MAX phase with the cohesive energy of the
competing equilibrium phases at corresponding composition as given by
the phase diagram in Fig. 6.2. All the calculations have been carried out
on stoichiometric phases without consideration to homogeneity ranges
(e. g. TiCx ; x=1 and Ti5 Si3 Cx ; x=0). Ti3 SiC2 is the only ternary phase
reported to exist in this system.
Figure 6.2: Phase diagram at 1250◦ C for the Ti-Si-C system [62]
6.2. PHASE STABILITY IN TI-SI-C SYSTEM
41
The phase diagram shows that the competing phases for Ti 4 SiC3 ,
Ti5 SiC4 and Ti7 Si2 C5 are Ti3 SiC2 and TiC, while the competing phases
for Ti2 SiC and Ti5 Si2 C3 are Ti3 SiC2 , TiSi2 and Ti5 Si3 (in reality Ti5 Si3 Cx ).
Table 6.2 lists the cohesive energy per atom (Ecoh) for each of the calculated MAX phases and for the competing equilibrium phases. The
stability of a given MAX phase is determined by the energy difference
(∆E) between the MAX phase and its competing phases. A negative
∆E indicates a stable phase. A positive ∆E suggests that it will decompose or not formed at all in favor for the competing phases.
From Table 6.2, it can be seen that there is a very small, but negative, energy difference of -0.008 eV/atom, between the 211 phase and
its competing phases. This suggests that the 211 could be stable, but
it should be noted that the calculated energy difference is so small that
it is approaching the accuracy of the calculations. This phase may be
metastable and can be synthesized as thin films. A more significant negative energy difference is calculated for the 413 compound with −0.029
eV/atom. This suggests that the 413 phase actually is stable. For
the 514 compound, the calculated energy difference is clearly positive,
+0.037 eV/atom and this phase should therefore not be stable. The
calculations of the new MAX phase structures, 523 and 725, show that
the energy difference is positive compared with the competing phases;
+0.007 eV/atom for the 523 compound (again, this energy difference
approaches the accuracy of the calculations) and +0.030 eV/atom for
the 725 compound. This suggests that both the observed inter-grown
phases should not be stable. However, it should be noted that even
though ∆E > 0 the calculated energy differences are very small and
that the compound still may be formed under metastable conditions.
Table 6.2: Formation energy in Ti-Si-C system
Si/Ti
decompositions
Ti2 SiC
0.5
0.75Ti3 SiC2 +0.107TiSi2 +0.1429Ti5 Si3
Ti3 SiC2 0.33
0.75Ti3 SiC2 + 0.25 TiC
Ti4 SiC3 0.25
Ti5 SiC4
0.2
0.6Ti3 SiC2 + 0.4 TiC
0.4 0.9Ti3 SiC2 + 0.0429 TiSi2 + 0.057 Ti5 Si3
Ti5 Si2 C3
0.8571Ti3 SiC2 + 0.1429 TiC
Ti7 Si2 C5 0.286
∆E
-0.0076
-0.0288
0.0373
0.0359
0.0278
CHAPTER 6. MAX PHASES
42
6.3
Chemical bonding in 312 phases
Three types of bonding, namely, metallic, covalent and ionic contribute
to the bonding in MAX phases. The high electrical conductivity is
attributed to the metallic bonding parallel to the basal plane (Ti I and
TiII ) and high bulk modulus is attributed to the strong Ti-C covalent
bond. The calculated balanced crystal orbital overlap (BCOOP) [63]
is used as a tool to study the chemical bonding in the Ti-Si-C system.
BCOOP can easily indicate the covalent bonding between two types of
orbitals. In Ti3 SiC2 , we are interested in the bonding between α 1 : Ti
3d states and α2 : C 2p states
BCOOPα1 ,α2 =
α1 |α2 δ( − n (k)) α α |α
n,k
(6.1)
Here n (k) is the Kohn-Sham eigenvalue, each eigenvector can be de
composed into non-orthogonal contributions |n, k = α |α, and |α =
i ci (n, k) |i. The denominator in Eq. (6.1),
α α1 |α2 , is introduced
to balance the bonding and antibonding states. BCOOP has positive
value for bonding states and negative value for the antibonding states,
the intensity of peaks and areas gives an indication of the strength of
the bonding.
0.4
Ti I-C
Ti II-C
Ti II-A
0.2
0.0
Ti3SiC2
-0.2
-0.4
0.4
Ti3AlC2
0.2
BCOOP (1/eV)
0.0
-0.2
-0.4
0.4
Ti3GeC2
0.2
0.0
-0.2
-0.4
0.4
TiC
0.2
0.0
-0.2
-0.4
-6
-4
-2
0
2
4
6
E-EF (eV)
Figure 6.3: BCOOP for Ti3 SiC2 , Ti3 AlC2 , Ti3 GeC2 and TiC.
6.4. XAS AND XES CALCULATION
43
In Fig. 6.3, the area under the TiII -C peaks is larger than the area under TiI -C peaks indicates that TiII -C bond is stronger than TiI -C bond.
While TiII -Al BCOOP located closer to the fermi level compared with
TiII -Ge and TiII -Si suggests that TiII -Al bond is weaker than the other
two types of bonds. This can easily be understood from the electronic
configuration. Al has three valence electrons instead of four valence electrons in Si and Ge, which should form less strong covalent bond because
of less saturated bonds.
6.4
XAS and XES calculation
The X-ray absorption (XAS) and X-ray emission (XES) in paper XIV
has been calculated using WIEN2K [64] code. The electric-dipole approximation has been employed, which means that only the transitions
between the core states with angular momentum l to the l ± 1 components of the conduction bands have been considered.
In Paper XIV, we show the calculated XES spectra of Ti 3 SiC2 ,
Ti3 AlC2 and Ti3 GeC2 and TiC. The calculated 3d DOS was projected
by 3d, 4s → 2p dipole transition matrix elements. We may notice that
Ti3 AlC2 shows a much more pronounced double peak structure compared with Ti3 SiC2 and Ti3 GeC2 (Fig. 6.4).
There is an extra peak located at -1 eV below the fermi level for
Ti3 AlC2 . To understand the orgin of this special double-structure in
Ti3 AlC2 , we plot the band structure for these three phases (as shown
in Fig. 6.5). A flat band between L to M, and K to H around -1 eV
corresponds to a high density of state in DOS, and this originates from
Si p state. The extra peak at -1 eV in Fig. 6.4 could due to the Ti-Si
hybridization. The other flat band between K to H symmetry points
attributes to the higher DOS of Ti at EF in Ti3 AlC2 .
6.5
DOS with electrical conductivity
The density of states at Fermi level N(E F ) in the simplest approximation
is directly related to the electrical conductivity. Our calculated N(E F )
for 211, 312, 413 and 514 phases are 0.36, 0.33, 0.29 and 0.25 states/eV
per atom, respectively. This can be compared to TiC, which has a
pseudo gap at the Fermi level with N(EF ) close to 0.1 states/eV per
atom, showing only weak metallic behavior. A trend of decreasing N(E F )
in the sequence of 211, 312, 413 and 514 compounds can be seen as
decreasing metallicity of MAX phases with increasing number of n. A
calculated N(EF ) = 3.96 states/eV per unit cell in 312 is in very good
CHAPTER 6. MAX PHASES
44
L3
Ti I + Ti II
Ti3SiC2
Ti3AlC2
Ti3GeC2
TiC
L2
Intensity (arb. units)
-20
-15
-10
-5
0
5
10
-5
0
5
10
-5
0
5
10
E-EF (eV)
Ti II
-20
-15
-10
-15
-10
E-EF (eV)
Ti I
-20
E-EF (eV)
Figure 6.4: Calculated Ti L edge XES spectra of Ti 3 SiC2 , Ti3 AlC2 and
Ti3 GeC2 and TiC for TiI , TiII and the sum.
2
1
0
-1
Energy (eV)
-2
2
1
0
-1
-2
2
1
0
-1
-2
ΓA
LM
Γ
KH
A
Figure 6.5: Band structures of Ti3 SiC2 , Ti3 AlC2 and Ti3 GeC2 .
6.5. DOS WITH ELECTRICAL CONDUCTIVITY
45
4
DOS (eV/states unit cell)
Ti3SiC2
TiI
TiII
3
2
1
0
-15
-10
-5
0
5
10
15
E-EF (eV)
Figure 6.6: TiI and TiII type of DOS in Ti3 SiC2
agreement with experimental data of 4.42 states/eV per unit cell [66]
derived from heat capacity data.
All Tin+1 SiCn (n=1 to 4) MAX-phase films show excellent conducting properties (see Table 6.3). This is in good agreement with the DFT
calculations, which show a relatively high DOS at Fermi level for all
these compounds. The electrical conductivity properties of Ti n+1 SiCn
should approach TiC with increasing number of Ti layers. Zhou et al
[65] suggest that the difference in charge density distribution between
TiI and TiII layers indicate that the TiII layers contribute more to the
electrical conductivity than TiI layers. We also observe this from different contribution to DOS at EF from TiI and TiII , as shown in Fig. 6.6
for Ti3 SiC2 .
Therefore, the number of TiII layers per unit cell plays an important
role in the electrical conductivity. The highest calculated N(E F ) in the
Tin+1 SiCn system suggests that 211 has higher conductivity than the
other MAX phases, which is also consistent with that it only contains
TiII type. As shown in Table 6.3, the ratio of TiII /Ti in one unit cell are
Table 6.3: Density of state at fermi level (N EF ) and electrical resistivity.
TiII /Ti
TiC
Ti2 SiC
Ti3 SiC2
Ti4 SiC3
Ti5 SiC4
1
0.667
0.5
0.4
N(EF )
0.1
0.36
0.33
0.29
0.25
Resistivity(µΩcm)
200-260
25-30
50
-
CHAPTER 6. MAX PHASES
46
1 (4/4), 0.667 (4/6), 0.5 (4/8), 0.4 (4/10) in the sequence of 211, 312,
413 and 514, suggesting a decrease of the electrical conductivity with
increasing number of n.
6.6
Optical properties
In metals, optical absorption occurs through two processes. One is the
intraband transition, the excitation of the electrons at the fermi level.
The Drude term represents a phenomenological way to describe the intraband transition, which contributes at low energies [67]. The second
process is interband transitions, i.e. the excitation of electrons from an
occupied band to an empty band. These two contributions combined to
give the total optical response.
Drude term is composed of two parts: relaxation time and plasma
frequency.
1) Relaxation time. This plays a fundamental role in the theory of
metallic conduction. We pick an electron randomly in the free electron
gas, on the average, it travels for a time τ before its next collision. The
relaxation time can be estimated via the resistivities ρ and electron denm
−14 to 10−15 sec
sity n, τ = ρne
2 . At room temperature, τ is typically 10
[68]. Although there is no formally correlation between the relaxation
times characterizes the (longitudinal) electrical conductivity and relaxation obtained in the Drude expression (describing transverse response),
we believe that these two relaxation time are strongly correlated to each
other. The inverse of τ , i.e. the collision frequency, often given the
symbol Γ is a measure of the broadening of the the phenomenological
Drude feature.
2) Plasma frequency. For the free electron gas ofdensity n, the os2
cillations of the electron plasma have frequency, ω p = 4πne
m . The ωp is
the frequency where the real part of the dielectric function goes through
zero from below, and the imaginary part approaches zero from above.
A sharp reflectance drop at ωp is a characteristic for high conductance
metal.
It is known TiN has a high reflectivity in the infrared, and a low
reflectivity for shorter wavelengths. It is this property makes TiN a
promising material for solar energy applications. The low reflectance
in the region of blue and violet light (2.8-3.5 eV) for TiN renders it
goldlike color [70]. With Al addition to TiN, the reflectivity decreases
dramatically in the infrared region and increase in the visible light region
(as shown in Fig. 6.7). The decrease of the reflectivity in the infrared
region is because of lower plasma frequency. The dip shifts from 2.8
6.6. OPTICAL PROPERTIES
47
TiC
TiN
Ti3SiC2
Ti4AlN3
0.8
REFLECTIVITY
0.6
0.4
0.2
0
1.5
2
2.5
3
3.5
4
4.5
ENERGY (eV)
Figure 6.7: Reflectivity for Ti3 SiC2 and Ti4 AlN3 . The experimental
data for TiC and TiN are taken from Fuentes et al. [69].
eV to 3.7 eV, i.e. outside the region of visible light. This shift is not
good for thermal solar collector applications, which require high solar
absorption, i.e. low reflectance in the range 1.5-3.5 eV. To minimize radioactive losses the collector surface should simultaneously exhibit low
thermal emittance at the working temperature, i.e. high reflectance in
the infrared region. As just mentioned the alloying of aluminum into
TiN annihilates both of these optical advantages, but with maintained
mechanical strength as well as thermal and chemical inertness. This
point makes another technical application for Ti 4 AlN3 . The reflectivity
minimum 0.3 (30%) for Ti4 AlN3 is higher than 0.2 (20%) for pure TiN.
It can therefore be used to avoid solar heating on space-crafts and also
increase the radioactive cooling due to the increased thermal emittance
compared to TiN [71]. The MAX-phases are therefore candidate materials for coatings in future space-missions to Mercury. The reflectance
of Ti3 SiC2 shows similar behavior with TiC, with almost constant value
in the region we have studied. The constant reflectance in the visible
light region makes Ti3 SiC2 look metallic grey.
It is known that in TiC the interband band transition occurs at
low energies even less than 0.1 eV [72], there is almost no free-electron
like region. In contrast, TiN exhibits a free-electron like region, below
2.5 eV, with high and constant reflectance [73]. Early band structure
calculations by Neckel et al. [74] explained different onset of transition
energy for TiC and TiN. The position where E F situated is the key
to the onset of interband transition energy or plasma energy. In TiN,
EF is located at the metal Ti-d dominated states, the free-electron like
behavior comes from the d electron-gas. While in TiC, C has one electron
less than N, EF moves down to the minimum of DOS, it becomes semi-
CHAPTER 6. MAX PHASES
48
30
25
Ti3SiC2
C-p
Ti-d
DOS (States/eV unit cell)
20
15
10
5
0
30
Ti4AlN3
N-p
Ti-d
25
20
15
10
5
0
-20
-15
-10
0
-5
Energy (eV)
5
10
15
Figure 6.8: Density of states for Ti3 SiC2 and Ti4 AlN3 .
metal. The plasma energy is proportional to the position of E F above
the DOS minimum. In Fig. 6.8, we plot the density of states for Ti 3 SiC2
and Ti4 AlN3 , We can observe that as Si addition to TiC, E F moves up
to higher energy [75]; while as Al addition to TiN, E F moves down to
the bottom of DOS valley [76]. This indicates that Ti 3 SiC2 should have
higher plasma energy also higher conductivity than Ti 4 AlN3 .
6.7
Surface energy
Surface energy is one of the fundamental properties in the surface science. In the Ti-Si-C system, surface energy calculations for the (0001)
surface have been made. The surface energy for TiC (111) was used as
a reference to compare the the Ti-C bond strength between TiC and
the Ti-Si-C system. Because MAX phases have the layered hcp structure along the [111] direction of the TiC, if we stack fcc TiC along [111]
direction, and replace a single layer of Si with C, we will get the hcp
Ti3 SiC2 structure. Due to different bond strength between Ti-C and
Ti-Si bond, the atom positions are slightly rearranged along the z direction in Ti3 SiC2 compared with TiC. In order to study the (0001) surface
energy of the Ti-Si-C system, it would be very important to look at the
surface energy in TiC.
Firstly, we compared the calculated surface energy for the unpolar
(001) surface of TiC with other existing results. The basis vector for fcc
was represented using a primitive tetragonal basis (1/2, -1/2, 0) (1/2,
1/2, 0) (0 0 1). The resulting surface energy 2.26 J/m 2 agrees well
with 2.254 J/m2 reported by Arya et al. [77], which shows that our
calculations should be reliable.
The Ti terminated surface has the lowest energy for the polar TiC
6.7. SURFACE ENERGY
49
(111) surface, while the C surface terminations yield much high surface
energies. A calculated Ti terminated (111) surface energy E f = 3.122
J/m2 was reported by Arya et al. [77]. The anisotropy in fcc TiC can be
described by the ratio γ100 /γ111 [78], a ratio of 1 gives a perfect isotropic
structure. A ratio 1.875 for TiC indicates that it is strongly anisotropic.
In order to study the Ti-C bond strength in TiC, we take the Ti and C
terminated surface. The calculated average surface energy 5.4 J/m 2 is
much higher than the result of Arya et al., which indicates that the C
terminated surface might have even higher surface energy.
The (0001) surface energy in the Ti-Si-C system was calculated by
inserting vacuum layers with the thickness 12 atomic layers (1 lattice parameter along z direction). As we know, there are three types of bonding
TiI -C, TiII -C and Ti-Si, we insert the vacuum at different position to
break the different types of bonds. The average of all these three types
of surface energy is the average surface energy of the (0001) plane,
E=
Evac − Ebulk
.
2
(6.2)
where Evac is the total energy of surface slab plus vacuum layers and
Ebulk is the bulk energy. To get reliable surface energies, we have to
converge our surface energies with respect to k-points and vacuum layers.
In this calculation, we have checked convergence up to 24 vacuum layers.
The average surface energies are shown in Table 6.4. Our calculated
average surface energy is strongly correlated to the bond strength. The
relation between bulk modulus and the bond strength is illustrated, one
can see from Table 6.4, the bulk modulus increases from 205 to 245 with
increasing bond strength. But all these MAX phases is softer than TiC
because of the much weaker Ti-Si bond. The Ti-C (both Ti II -C and
TiI -C) bond is calculated to be much stronger than the Ti-Si bond. The
TiII -C type bond is slightly stronger than the Ti I -C type bond, and the
TiI -C bond strength is similar to the Ti-C bond in TiC.
Table 6.4: The surface energy (eV) and bulk modulus (GPa) in the
Ti-Si-C system.
Ti-Si
TiC
211
312
413
1.593
1.585
1.583
TiI -C
3.052
3.083
3.317
TiII -C
3.330
3.583
3.586
γs
3.052
2.462
2.750
2.951
B (GPa)
289
205
233
245
CHAPTER 6. MAX PHASES
50
6.8
Ductility
Crystalline materials divide into two categories depending on their number of slip planes. Materials have a large multiplicity of easy glide slip
demonstrate significant ductility, and the dislocation activity in this system is irreversible. Meanwhile, materials that exhibit limited amounts
of slips are generally brittle (e.g. ceramics) [79], and constrain their use
in a number of technological applications.
One criterion proposed by Paugh for isotropic materials is the ratio between B (bulk) and G (shear modulus). There B represents the
resistance to fracture, while G represents the resistance to plastic deformation. The critical value around 1.75 differentiates ductile from brittle
materials [80].
An approximation to calculate the elastic properties for cubic structure from Grüneisen constant was introduced by Korzhavyi et al. [81].
We take the simple expression for the determination:
γ = −1 −
V ∂ 3 E/∂V 3
2 ∂ 2 E/∂V 2
(6.3)
E is the total energy, which can take the integrated form of the third
order BM-EOS,
V3
V
7/3
9
B0 [(4 − B0 ) V02 − (14 − 3B0 ) V04/3
E(V ) = − 16
V
5/3
+(16 − 3B0 ) V02/3 ] + E0
(6.4)
Then we can derived the γ at V = V0
1
1
γ = B − ,
2
2
(6.5)
This γ is exactly the Dugdale-MacDonald Grüneisen constant, which
is proved to lie closest to the that determined from ab initio calculation
[82].
The relation between the Grüneisen constant and Poisson’s ratio is
σ=
4γ − 3
6γ + 3
(6.6)
The ratio between B and G can be determined through Poisson’s
ratio
2(1 + σ)
.
(6.7)
B/G =
3(1 − 2σ)
Our calculated Poisson’s ratio is in very good agreement with the
ab initio value 0.25 calculated by Holm et al. [83] and in reasonable
6.8. DUCTILITY
51
Table 6.5: Comparison between TiC, Ti2 SiC, Ti3 SiC2 and Ti4 SiC3 . for
bulk modulus (B0 ), first pressure derivative of bulk modulus (B 0 ), shear
modulus (G), Young’s modulus (E) and Poisson’s ratio (σ).
B0
B0
G
E
σ
B/G
TiC
289
3.892
183
454
0.238
1.578
211
205
4.043
120
302
0.254
1.703
312
233
4.045
137
343
0.255
1.704
413
245
4.069
142
357
0.257
1.725
agreement with the experimental value 0.20 [84]. In Table 6.5, we list
the B/G ratio in the sequence of TiC, Ti2 SiC, Ti3 SiC2 and Ti4 SiC3 .
It is well known that TiC is quite brittle. We can observe the pronounced change of B/G with Si addition to TiC and this makes it become more ductile. However, in our calculation, the B/G is very sensitive to the fitting parameter B and the slight increase or decrease of
B can affect the ratio. Therefore to get reliable values for B/G, elastic
constants calculations have to be carried out for this system.
52
CHAPTER 6. MAX PHASES
Chapter 7
Sammanfattning på svenska
Walter Kohn fick nobelpris i kemi 1998 för sina insatser inom elektronstrukturteorin. Kohns vetenskapliga arbeten har gjort det möjligt för
kemister och fysiker att beräkna fundamentala egenskaper hos molekyler
och fasta ämnen med hjälp av datorer, utan att behöva göra några som
helst experiment i laboratoriet.
Redan under 1930-talet var fysiker i full fart med att utarbeta de
kvantmekaniska ekvationer som bestämmer uppförandet hos ett system
med många elektroner. Tyvärr kunde de inte lösa dessa ekvationer utom
för de allra enklaste fallen. De teoretiska fysikerna utvecklade en mängd
approximationsmetoder, men ingen av dem var tillräckligt slagkraftig
eller tillräckligt enkel att genomföra. 1964 bevisade Kohn tillsammans
med Pierre Hohenberg en hypotes som sedan blev central för deras
lösning till elektronproblemet, något som idag kallas för densitetsfunktionalteori (DFT). DFT skiljer sig från de traditionella kvantkemiska
metoderna och ger inte en korrelerad N-kroppars vågfunktion. Många
av de kemiska egenskaperna hos molekyler och de elektroniska egenskaperna hos fasta ämnen bestäms av elektroner som växelverkar mellan
varandra och med atomkärnorna. Enligt en tidig version av DFT som
Kohn utarbetade tillsammans med Sham var det tillräckligt att känna
till medeltätheten för elektronerna i rummets alla punkter för att entydigt bestämma den totala energin, och därigenom en mängd andra egenskaper hos systemet. DFT-teorin är i grunden en s.k. en-elektronteori
och uppvisar många likheter med Hartree-Fock-teorin. DFT har kommit att bli den dominerande metoden under det sista årtiondet, varande
den som är potentiellt mest kapabel till att leverera noggranna resultat
till låga kostnader.
Med hjälp av den allra senaste formuleringen av densitetsfunktionalteorin har stora framsteg gjorts vad beträffar beräkning av mekaniska
och elektroniska egenskaper hos fasta ämnen. Olika approximationer
53
54
CHAPTER 7. SAMMANFATTNING PÅ SVENSKA
vad beträffar behandlingen av de utbytes-korrelationfunktionaler, som
används inom densitetsfunktionalteorin för grundtillstånd och exciterade tillstånd har lagts fram, exempelvis lokaldensitetsapproximationen
(LDA), generaliserande gradient approximationen (GGA), LDA+U och
GW-approximationen. Mängder av tillämpningar inom materialmodellering i allmänhet liksom för nanotuber, kvantprickar och molekyler
(även artificiella) har genomförts. Eftersom teorin har utvecklats mycket snabbt och nått en sådan avancerad nivå har tiden nu har blivit
mogen för ett teoribaserat angrepp vad beträffar support och supplement till experiment. Denna utveckling har också fått stor hjälp genom
den fortsatta förbättring av kraftfulla datorer som har gjort det möjligt
att beräkna materialegenskaper med en imponerande noggrannhet. Huvudtemat i denna avhandling är att använda mycket noggranna teoretiska ab initio-metoder för att förutsäga materialegenskaper och söka
efter nya teknologiskt användbara material.
Nyligen har monolagrade ternära föreningar M N+1 AXN , där N=1,2
och 3, M en tidig övergångsmetall, A är ett element från A-gruppen
(huvudsakligen III A och IV A), och X är antingen kol och/eller kväve,
dragit till sig ett alltmer ökat intresse på grund av deras unika egenskaper. De temära karbiderna och nitriderna kombinerar egenskaper hos
både metaller och keramer. Liksom metallerna är de goda termiska och
elektriska ledare med elektriska och termiska konduktiviteter varierande
från 0.5 till 14×106 Ω−1 m−1 , och från 10 till 40W/m·K, respektive. De
är relativt mjuka med en Vickers hårdhet på omkring 2-5 GPa. På
samma sätt som keramer är de elastiskt styva. Några av dem såsom
Ti3 SiC2 , Ti3 AlC2 och Ti4 AlN3 uppvisar även excellenta mekaniska egenskaper vid höga temperaturer. De motstår termisk chock och är ovanligt
tåliga gentemot skador, och uppvisar utmärkt korrosionsbeständighet.
Framför allt, till skillnad från konventionella karbider och nitrider, kan
de formbehandlas med konventionella verktyg utan smörjmedel, vilket
har stor praktisk betydelse för användandet av MAX-faserna. De ovan
nämnda egenskaperna gör MAX-faserna till en ny familj av teknologiskt
viktiga material. Systematiska studier av MAX-faserna vad beträffar
elektronstruktur, bindning, elastiska och optiska egenskaper har utförts
i denna avhandling. Elektronstrukturen och bindningsegenskaper hos
dessa fasta ämnen är nyckeln till att exploatera egenskaperna hos MAXfaserna.
Avhandlingen är uppdelad enligt följande. Sektion 2 beskriver huvuddragen hos densitetsfunktionalteorin. Sektion 3 beskriver de beräkni
ngsmetoder som jag har använt för mina beräkningar. Mina forskningsresultat är indelade i tre delar:
1) Fasövergångar och relaterade egenskaper såsom fasstabilitet och
55
tillståndsekvationer, vilket behandlas i kapitel 3.
2) Beräkning av linjära optiska egenskaper hos vissa halvledare, såsom
solcellsmaterialen CuIn(Ga)Se2 och den scintillerande kristallen PbWO 4 .
3) I kapitel 5 behandlas tillämpningar av densitetsfunktionalteorin på
MAX - faserna, inkluderande deras elektroniska, bindnings - , mekaniska
och optiska egenskaper.
56
CHAPTER 7. SAMMANFATTNING PÅ SVENSKA
Acknowledgments
The work presented above was made possible by support of the Condensed Matter Theory Group (Fysik IV) at the Uppsala University (UU)
under the supervision of Docent Rajeev Ahuja. Many other colleagues
in the group have greatly contributed to my work and enriched my life.
Words are not enough to express my cherishment of all the memorable
moments of my stay in Fysik IV. I would like to extend my sincere gratitude to all those that had given me a helpful hand in the past four years.
Among them, I am especially grateful to:
Docent Rajeev Ahuja, my supervisor, for your indispensable hand-tohand education and visionary supervision; for all of the encouragements
and supports you gave me, on my research work, on my publication
and thesis write-up. I am especially thankful to you for being the one
recruited me to the group.
Prof. Börje Johansson, for your patience in revising my papers and
thesis. Thank you especially for treating me as a friend, enlightening
me with your wisdom in academics as well as in life.
Prof. Olle Eriksson, for your interesting lectures, kind jokes, and for
the pleasant working experience.
My great thanks will also be directed to my co-workers J. M. Wills,
Anna Delin and Kay Dewhurst who implemented the code, which fundamentally helped my work.
Prof. Ulf Jansson, Prof. C-G Ribbing, Prof. Michel Barsoum, Prof.
Jochen M. Schneider, Dr. Zhimei Sun, Dr. Martin Magnuson and Dr.
Jens -P. Palmquist, I appreciate your collaborations, which significantly
enriched my knowledge on MAX phases.
Dr. Yi Wang, for your kind helps in solving the problems I encountered during the work. Ms. Lunmei Huang and all my Chinese friends
in Uppsala for your sharing the feelings of living in foreign country in
our mother language.
Jorge Osorio, Alexei Grechnev, Weine Olovsson, Jailton Souza de
Almeida, Erik Holmström and all other colleagues in the Condensed
Matter Theory Group in Uppsala University, for your generous assistance in my study, my computer problems and everything else. Most
57
58
CHAPTER 7. SAMMANFATTNING PÅ SVENSKA
importantly, for your care-free fellowships!
Sincere gratitude is to my dear parents, and Enyin, Lingli, Siyun as
well as all of other family members, for your love, your deep-touching
love!
Lastly, Yupeng, for your dearest love! Life would not be the same
as meaningful with you out.
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Acta Universitatis Upsaliensis
Comprehensive Summaries of Uppsala Dissertations
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