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DOMENICO DI SANTE
Density Functional Theory and
Group Theoretical Analysis in
the Study of Hydrogen Bonded
Organic Ferroelectrics
TESI DI LAUREA MAGISTRALE IN
FISICA
Relatore: Prof. Alessandra Continenza
Co-relatore: Dr. Silvia Picozzi
Co-relatore: Dr. Alessandro Stroppa
Università degli Studi dell’Aquila
Facoltà di Scienze MM. FF. NN.
Dipartimento di Fisica
Luglio 2011
Domenico Di Sante: Density Functional Theory and Group Theoretical
Analysis in the Study of Hydrogen Bonded Organic Ferroelectrics, Tesi
di Laurea Magistrale in Fisica, © luglio 2011.
It’s easier to leave than to be left behind.
— REM (leaving New York)
Dedicato a tutte le persone a me care.
A Chiara ed a tutta la mia famiglia.
ACKNOWLEDGEMENTS
I want to acknowledge for their fundamental support during
the last two years Dr. Silvia Picozzi and Dr. Alessandro Stroppa.
Without their help, this work would never see the light. A special
thank to Prof. Alessandra Continenza.
With all my heart, I thank Chiara - my life - for everything, and
always I will do it.
I acknowledge all my family, Francesco, Franca, Simone, Maria,
Giovanni and Giancarlo, and my best friend Beatrice for their
wonderful presence and moral support. Special acknowledgments
also to Daniele, Paolo and to all my friends.
iv
ABSTRACT
This thesis tackles structural and electronic properties of a series of new hydrogen bonded organic ferroelectrics and of a manganese based metal-organic framework (MOF) from an ab initio
point of view. Symmetry related materials properties such as
symmetry mode analysis of polar distortions are then investigated through group theoretical methods. When possible, comparisons with experimental results are reported.
Relatori:
Prof. Alessandra Continenza
University of L’Aquila
....................................................
Dr. Silvia Picozzi
CNR-SPIN, L’Aquila
....................................................
Dr. Alessandro Stroppa
CNR-SPIN, L’Aquila
....................................................
Candidato:
Domenico Di Sante
University of L’Aquila
....................................................
CONTENTS
Introduction
I
1
Theoretical and Computational Methods
1
THE
1.1
1.2
1.3
1.4
1.5
1.6
DENSITY FUNCTIONAL THEORY
6
The Many-Body Problem
6
The Hohenberg-Kohn Theorem
8
The Kohn-Sham Equations
9
The Exchange-Correlation Term
11
Exact Properties for Exc [n]
14
Local Spin Density Approximation
16
2
DFT
2.1
2.2
2.3
2.4
2.5
2.6
CALCULATIONS
18
PBE functional
19
Hybrid functionals
23
The Electronic Ground State
27
Optimization of Atomic Positions
29
The Modern Theory of Polarization
31
The VASP code
33
3
SYMMETRY ANALYSIS
35
3.1 The Pseudo Tool
35
3.2 The Amplimodes Tool
II
4
5
36
Computational Results and Analysis
POLAR DISTORTIONS IN H-BONDED
ORGANIC FERROELECTRICS
39
4.1 Structural Properties
40
4.2 The Ferroelectric Polarization
46
4.3 Symmetry-Mode Analysis of Ferroelectricity
4.4 Conclusions
60
vi
38
51
CONTENTS
5
POST-DFT STUDY OF CROCONIC ACID PROPERTIES
5.1 Structural Properties
63
5.2 Electronic Properties
66
6
MULTIFERROICITY IN A MANGANESE BASED MOF
71
6.0.1 Different types of multiferroics
72
6.0.2 Metal-organic frameworks 72
6.1 Crystal Structure and Spin Ordering of Mn-MOF
74
6.2 Microscopic Origin of the Spontaneous Polarization
78
6.3 Electronic Properties
83
7
CONCLUSIONS
63
86
III Appendix
90
a ABOUT CORRELATIONS
b
THE PAW METHOD
91
96
c A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
BIBLIOGRAPHY
109
102
vii
INTRODUCTION
A material showing a spontaneous electric polarization that
can be reversed by the application of an external electric field
is said to be ferroelectric. Ferroelectricity has long been an important topic in condensed-matter with important applications
in memory devices [1]. The connection between ferroelectricity and organics was established in 1920 with the discovery of
the Rochelle salt, the first ferroelectric crystal based on organic
molecules [2]. Nevertheless, examples of organic ferroelectrics
have not been so abundant in the last decades, despite the fact
that due to their lightness, flexibility and non-toxicity, they may
find many new applications in the emerging field of organic electronics.
Ferroelectrics appear in the form of either solid (crystalline or
polymeric) or liquid crystals, where the electric polarization P
as function of the field strength E draws a hysteresis curve between opposite polarities. The critical electric field necessary to
reverse the polarization is known as coercive filed. The electric
bi-stability can be used, for example, in the development of ferroelectric random access memories (FeRAMs) and ferroelectric
field-effect transistors [3]. Ferroelectric compounds show a Curie
temperature Tc for the paraelectric-ferroelectric phase transition:
as the temperature approaches Tc , the dielectric constant ε, which
obeys the Curie-Weiss law, reaches large values to be used high-ε
condensers and capacitors. The other important property from
the technological point of view is pyroelectricity, e.g. a temperature dependence of the spontaneous polarization generates an
electric current when both ends of the polarized ferroelectric are
short-circuited. Just below Tc the pyroelectric effect becomes especially large, a feature very useful for thermal-image sensors
and infrared detectors. Furthermore, ferroelectricity establishes
a sort of bridge between electric and mechanical properties; the
stress generates electric polarization, whereas the electric field
creates strain in the material. Electrostriction and piezoelectric effects are used in actuators, transducers, ultrasonic motors, piezo-
1
CONTENTS
Figure 1: Schematic representation of two hydrogen bonded organic
molecules. In green, red and black, respectively, carbon, oxygen and hydrogen atoms are shown. Grey thin solid lines refer
to hydrogen bonds.
electric elements and microsensors, just to cite some examples. In
addition, the polar crystal structure yields second-order optical
nonlinearity, causing second-harmonic generation activity and a
linear electro-optic effect, useful in technological areas like electronics, electro-optics and electromechanics. Most of these operations under ambient conditions need a Tc near or above room
temperature.
New approaches for materials design of organic ferroelectrics
have been recently developed [4]. In particular, a special class of
materials is that of hydrogen bonded organic ferroelectrics (see
Figure 1) , in which dynamic protons in O-H· · · O units (where
O-H denotes a covalent bond and H· · · O a hydrogen bond) trigger the ferroelectric ordering of the lattice; as will be explained
in closer detail in this thesis, a collective site-to-site transfer of
protons in the O-H· · · O bonds switches the spontaneous polarization. During the last few years, great work has been done in
materials design for this kind of ferroelectrics, mainly because
2
CONTENTS
small coercive fields - an important feature in the realization of
electronic devices - are required to reverse the polarization.
The realization of ever smaller tunable devices is a major challenge in nanoelectronics; as a result, considerable efforts have
been devoted to multifunctional materials in the last few years.
Among them, the porous crystalline materials known as metalorganic frameworks (MOFs) are currently a hot topic of research,
especially for their multiferroic properties, i.e. the ability to host
both a magnetic and a ferroelectric order at the same time. The
exciting features of this new class of materials for device applications come from their hybrid nature, benefiting from the characteristics of both the inorganic and organic building blocks. Such
hybrid nanoporous structures in which metal ions are embedded
in an organic framework have not been considered for multiferroic purposes until recently [5]. Furthermore, the presence of organic molecules in the structures allows hydrogen bonds to form
between the MOF’s components and to play an important role
in the stabilization of a ferroelectric state. For example, multiferroic properties can be employed to electrically control magnetic
memories or in magnetoelectric sensors.
This thesis tackles structural and electronic properties of a series of new hydrogen bonded organic ferroelectrics and of a manganese based metal-organic framework (MOF) from an ab initio
point of view. In order to determine materials properties, density
functional theory (DFT)-based calculations have been performed
at different levels of approximation for the exchange-correlation
functional. DFT replaces the complicated many-body problem
of interacting electrons with a simpler one that requires only the
knowledge of the distribution of electron charge in space. The
foundations of the theory were set by Pierre Hohenberg and Walter Kohn [6] in 1964. Symmetry related materials properties such
as symmetry mode analysis of polar distortions are then investigated through group theoretical methods as implemented in the
Bilbao Crystallographic Server [7]. When possible, comparisons
with experimental results are reported.
The work is organized as follows: in part I Density Functional
Theory, computational methods and group theory methodologies
used in our study are illustrated, whereas part II contains a full
description of results.
3
CONTENTS
Results regarding the organic ferroelectrics CBDC and PhMDA (chapter 4) will be presented in a paper accepted by Physical Review B, and will be illustrated in the Psi-k/CECAM/CCP9
Biennial Graduate School in Electronic-Structure Methods 2011 in Oxford. Furthermore, the abstract will be submitted for the XCVII
Congresso Nazionale della Società Italiana di Fisica that will take
place in L’Aquila from 26 to 30 September 2011. Results regarding the organic ferroelectric croconic acid (chapters 4 and 5),
and regarding the manganese-base metal-organic framework
(chapter 6), will be presented in forthcoming publications.
4
Part I
Theoretical and
Computational Methods
5
1
THE DENSITY FUNCTIONAL
THEORY
The key point of condensed matter physics is to investigate the
properties of solids, and one way to do that is to calculate their
electronic structure. The knowledge of the electronic structure of
solids is not only helpful for understanding and interpreting experiments, but it also allows the prediction of properties of newly
designed materials. Density Functional Theory is currently one
of the most useful methods for investigations in this field.
DFT is a ground state theory where the electronic charge density the relevant physical quantity, through which all other fundamental quantities can be calculated. It is a matter of fact that DFT
well describes structural and electronic properties for a wide class
of materials: from simpler atoms and molecules to crystalline
structures, up to complex extended systems like liquids. Furthermore, DFT is computationally quite simple. For these reasons it
has become a common tool in first-principles calculations aimed
at describing – or even predicting – properties of molecular and
condensed matter systems.
1.1
THE MANY-BODY PROBLEM
In order to determine the properties of a system, one needs to
solve the Schrödinger equation
ĤΨ = EΨ ,
(1.1)
where Ĥ is the Hamiltonian describing all the interactions of the
system, E is the total energy of the system, and Ψ is the manybody wave function containing all the information that can be
6
1.1 THE MANY-BODY PROBLEM
obtained about the nuclei and electrons in the system. The full
Hamiltonian can be expressed as
Ĥ = T̂e + T̂n + V̂ee + V̂nn + V̂en = −
h 2 X 2 X h2 2
∇
∇i −
2me
2MI I
i
+
X Z I e2
1 X e2
1 X Z I Z J e2
+
−
2
|ri − rj | 2
|RI − RJ |
|ri − RI |
i6=j
I
,
(1.2)
iI
I6=J
where T̂e and T̂n refer to kinetic energies of electrons and nuclei respectively, and V̂ee , V̂nn , V̂en are the interaction potential
terms (with obvious significance of symbology); indices i and j
refer to the electrons, and indices I and J to the nuclei.
In the "fixed" lattice approximation, obtained by ignoring the
nuclear kinetic operator in the total Hamiltonian 1.2, we are left
with the so-called electronic adiabatic Hamiltonian Ĥe (r; R) given
by (r and R are shorthand notations)
Ĥe (r; R) = T̂e + V̂ee + V̂nn + V̂en = T̂e + V̂(r; R)
.
(1.3)
In this Hamiltonian the variables R appear simply as parameters (instead of quantum dynamical observables); thus Ĥe (r; R)
belongs to the class of parameter dependent operators, which we
will discuss later in the context of Berry phase formalism for the
modern theory of macroscopic polarization in crystals.
We can take into account the nuclear kinetic operator in 1.2
only if we are interested in lattice dynamics, because of the large
difference in masses of electrons and nuclei, and this leads us to
the Born-Oppenheimer approximation [8].
The eigenvalues equation for the electronic Hamiltonian Ĥe (r; R)
is
Ĥe (r; R)Ψn (r; R) = En (R)Ψn (r; R)
,
(1.4)
where the electronic wavefunctions Ψn (r; R) and the eigenvalues En (R) depend on the parameters R; the suffix n summarizes the electronic quantum numbers. Even within the BornOppenheimer approximation, the number of particles (electrons)
entering the problem is so large and the electron-electron interaction so difficult to treat, that an exact solution is impossible.
Therefore, a number of further approximations must be applied
to make the problem solvable.
7
1.2 THE HOHENBERG-KOHN THEOREM
1.2
THE HOHENBERG-KOHN THEOREM
Within the interacting density functional theory, the complicated many-body problem of electrons is replaced by an equivalent but simpler problem of a single electron moving in an effective potential.
Let us consider a system of N interacting electrons (spinless for
the moment) under an external Coulomb potential V̂en for the
electrons-nuclei interaction. If the system has a non-degenerate
ground state, it is quite obvious that a unique ground state electronic density n(r) corresponds to the external potential. The
opposite is a less obvious result, that Hohenberg and Kohn established in 1964 [6]. The demonstration is simply based on the
fact that two different external potentials cannot have the same
ground state charge density. In other words, it is possible to
define the carghe density n(r) as a functional of the external potential, i.e. n[V(r)]. Suppose in fact that for a potential Vˆ 0 en
(such that V̂en − Vˆ 0 en 6= cost) eigenvalues equation 1.4 gives a
ground state wavefunction Ψ00 (r; R), and suppose for absurd that
n[V̂] = n[V̂ 0 ]; clearly the following relations must hold:
E00
0
0
= hΨ00 |Ĥe0 |Ψ00 i = hΨ00 |Ĥe + V̂en
− V̂en |Ψ00 i < hΨ0 |Ĥe + V̂en
Z
0
−V̂en |Ψ0 i = E0 + dr n(r)[V̂en
(r) − V̂en (r)] ;
(1.5)
by reversing the primed and unprimed quantities, one obtains an
absurd result, because the inequality in 1.5 is strict being Ψ0 and
Ψ00 eigenfunctions of different adiabatic electronic Hamiltonians.
A straightforward consequence of 1.5 is that the ground state
energy is uniquely determined by the ground state charge density, or, equivalently, the total energy of the system can be written
as a functional of the density:
Z
E[n] = dr n(r)V̂en (r) + F[n] ,
(1.6)
where F[n] is a universal functional of n(r) containing the kinetic
energy and the electron-electron interactions. For the functional
in 1.6, a variational principle holds: the ground state energy is
minimized by the ground state charge density. In this way, DFT
exactly reduces the N interacting particles problem to the determination of a function n(r) of the 3-coordinates which minimizes
8
1.3 THE KOHN-SHAM EQUATIONS
the functional in 1.6. Unfortunately, this is not known, and one
needs to adopt some schemes to obtain an expression for it.
1.3
THE KOHN-SHAM EQUATIONS
In 1965 Kohn and Sham [9] proposed to substitute the interacting many-particle potential V̂en (r) with an effective one-electron
potential V̂eff (r) holding the same ground state charge density.
The total energy functional 1.6 can be rewritten as
Z
E[n] =
dr n(r)V̂en (r) + Ts [n] + Exc [n]
ZZ
e2
n(r)n(r 0 )
+
drdr 0
,
(1.7)
2
|r − r 0 |
where the functional Ts [n] is the kinetic energy of the non interacting system and the last term, EHartree [n], represents the classical Coulomb interaction energy of an electronic cloud of density n(r). The term Exc [n] is the so-called exchange-correlation
functional, containing the exchange and correlation energies of
the interacting system and corrections to the kinetic energy term
that must be included passing from the one-electron to the manyparticle picture.
Minimizing equation 1.7 with respect to n(r), leads us to a
Scrödinger-like equation of the type
h2 2
−
∇ + V̂eff (r) Ψi (r) = εi Ψi (r) ,
(1.8)
2me
where εi and Ψi (r) are the so called Kohn-Sham eigenvalues and
eigenfunctions respectively. V̂eff (r) is the effective one-electron
potential associated to the same ground state charge density of
the interacting many-particle potential:
Z
n(r 0 )
V̂eff (r) = V̂en (r) + e2 dr 0
+ V̂xc (r) ,
(1.9)
|r − r 0 |
being the exchange-correlation potential given by the functional
derivative
∂Exc [n]
V̂xc (r) =
.
(1.10)
∂n(r)
9
1.3 THE KOHN-SHAM EQUATIONS
In terms of Kohn-Sham orbitals, the electron charge density
can be written as
occ
X
|Ψi (r)|2 ,
(1.11)
n(r) =
i
where the summation runs over all occupied one-electron KohnSham states.
In computations, electron density, Kohn-Sham eigenvalues and
eigenfunctions are calculated iteratively. The total energy of the
interacting system can at this point be written as
Z
occ
X
εi − EHartree [n] − dr n(r)V̂xc (r)
E[n] =
i
+Exc [n]
.
(1.12)
An overview on the resolution methodologies of equation 1.8
will be given later.
We just notice here that Kohn-Sham equations are standard
differential equations with a rigorously local effective potential
V̂xc (r); any difficulty in the solution procedure has been confined
to the choice of a reasonable guess of the exchange-correlation
functional Exc [n], known only in principle, as we shall see. Conceptually, the Kohn-Sham scheme determines exactly the electron
density and the ground state energy, however the eigenvalues εi
don’t have any physical meaning, since the Koopmans’ theorem
doesn’t hold for them. The identification of εi with occupied
and unoccupied one-electron states has to be justified. In general
experience shows that density functional theory tends to underestimate the energy band gap in semiconductors and insulators,
independently on the exchange-correlation functional is used in
1.9; however dispersion curves for valence and conduction bands
are well described.
Before concluding this section, we want to underline similarities and differences between Kohn-Sham and Hartree-Fock equations. Both can be derived from a variational principle, the minimization of the energy functional for the former case and the
minimization of the single Slater determinant energy for the latter. Furthermore, both can be resolved self-consistently. Differences come from the way electron-electron interactions are described; within the Kohn-Sham picture the exchange-correlation
10
1.4 THE EXCHANGE-CORRELATION TERM
functional appears, while in the Hartree-Fock equations the exchange term is taken into account exactly:
Z
X
Ψj (r)Ψ∗j (r 0 )
V̂xHF (r)Ψi (r) = −e2
δsi sj dr 0
Ψi (r 0 ) (1.13)
|r − r 0 |
j
where the summation over j extends only to states with parallel
spin direction. The correlation energy is traditionally defined as
the difference between the Hartree-Fock exchange energy and the
real energy. In density functional theory, the exchange-correlation
term cannot be calculated simply adding the correlation energy
term to the Hartree-Fock exchange, since it also include information regarding the many-particle kinetic energy of the interacting
system.
1.4
THE EXCHANGE-CORRELATION TERM
The Kohn-Sham theory is still incomplete until an explicit form
is given for the exchange-correlation functional. Many attempts
have been made to search a reasonable guess for it, however, the
problem is still open nowadays. Historically, the first suggestion
came from Kohn and Sham [9], who proposed the so-called local
density approximation, better known as LDA. They approximated
the functional Exc [n] with a function of the local density n(r) writing
Z
LDA
Exc [n] = dr n(r)εxc (n(r)) ,
(1.14)
where εxc (n(r)) is the exchange-correlation energy per particle
for a homogeneous electron gas (also known as jellium) with local density n(r). Many-body calculations performed using path
integral Monte Carlo methodologies [10] give accurate, and in
principle exact, results, which have been parametrized in several
ways [11, 12]. A simple analytical form by Perdew and Zunger
[13] is
√
εxc = −0.4582/rs − 0.1423/(1 + 1.0529 rs + 0.3334rs )
=
−0.4582/rs − 0.0480 + 0.0311 ln rs − 0.0116rs
+0.0020rs ln rs
(1.15)
11
1.4 THE EXCHANGE-CORRELATION TERM
respectively for rs 6 1 and rs > 1, where rs = (3/4πn)1/3 is a
function of the density. It is easily verified that the first term in
1.15 is just the Slater local approximation for the homogeneous
electron gas exchange energy:
3 2 3 1/3
[n(r)]1/3 .
(1.16)
εx = − e
4
π
Other parameterizations appeared in literature yield similar results, since they are very similar in the range of rs applicable to
solid-state calculations.
LDA, despite its simplicity, turns out to be much more successfully than expected, computationally less complicated and much
simpler than Hartree-Fock, yielding results of similar quality for
atoms, molecules and inhomogeneous systems, for which an approximation based on the homogeneous electron gas doesn’t look
to be appropriate. This success is not just an accident, but can be
partially explained looking for in more detail to the exchange correlation term. It can be shown [14] that the exchange-correlation
energy can be written as
Z
1
nxc (r, r 0 )
Exc =
drdr 0 n(r)
,
(1.17)
2
|r − r 0 |
where nxc (r, r 0 ) is the exchange-correlation hole, the charge missing
around a point r due to Pauli antisymmetry exchange effect and
Coulomb repulsion. In terms of the pair correlation function g(r, r 0 )
giving the probability to find an electron in r 0 if there is already
one in r, the exchange-correlation hole is defined as
Z1
0
0
nxc (r, r ) = n(r ) dλ[gλ (r, r 0 ) − 1] ,
(1.18)
0
(r, r 0 )
where gλ
is the pair correlation function for a system in
which the electron-electron interaction V̂ee is switched on adiabatically. It has been shown that for inhomogeneous systems,
LDA doesn’t give an accurate description for the whole exchangecorrelation hole function, but just for its spherical part. In the
lefthand side of Figure 2, the exchange hole nx (r, r 0 ) of an electron in a Ne atom is shown at distances r = 0.09 and r = 0.4 from
the nucleus. When the LDA results are compared with the exact
numerical ones, we notice that the two curves look very different.
12
1.4 THE EXCHANGE-CORRELATION TERM
Figure 2: Left: exchange hole nx (r, r 0 ) for a neon atom for two different
values of r, respectively (a) r = 0.09 and (b) r = 0.4 Bohr’s radii.
Right: spherical average of the neon exchange hole nx (r, r 0 )
times r 0 for the same values of r. The full curves give the
exact results and the dashed curves are obtained in the LDA
approximation. (From [15])
The exact results yield exchange hole densities which diverge at
the position of the nucleus; in contrast, the LDA exchange hole
densities reach their largest values at the position of the electron,
and the holes present larger extensions. Despite these considerable discrepancies, angular averages are pretty much the same in
both cases, as shown in the righthand side of Figure 2. One expects that similar conclusions hold for the exchange-correlation
hole. Looking at equation 1.17, one can easily see that it depends
only on the spherical part of nxc (r, r 0 ), and this explains at least
partially the good performances of LDA.
In treating materials with significant variations in the electronic
density, where for example directional bondings generate strong
gradients, LDA is a less good approximation. Furthermore, LDA
tends to underestimate by ∼ 40% energy band gaps in semiconductors and insulators, overestimating cohesive energies and
bond strengths. There have been several attempts to improve
upon and go beyond the local density approximation. Some of
13
1.5 EXACT PROPERTIES FOR Exc [n]
the problems of LDA can be avoid introducing gradient corrections, writing the exchange-correlation functional as function of
the local density and its gradient:
Z
EGGA
[n]
=
dr n(r)εxc (n(r), |∇n(r)|) ;
(1.19)
xc
this takes the name of generalized gradient approximation, usually
referred as GGA [16]. Gradient-corrected functionals yield in
general much better results than LDA, and open the way to theoretical studies of hydrogen bonded systems, such as water or
more complicated organic materials, providing descriptions of
Hydrogen bond’s properties not possible through simple LDA
based calculations. These reasons led us to perform almost all
simulations within the generalized gradient approximation for
the exchange-correlation functional, playing the Hydrogen bond
a crucial role in these organic ferroelectrics we have studied.
A deeper discussion about correlations is given in appendix A.
1.5
EXACT PROPERTIES FOR E xc [n]
As we have pointed out, the correlation term Ec [n] is a very
complicated object, and DFT would be useless if its exact knowledge is required. However, the practical advantage of writing
E[n] as in 1.7 is that the unknown term Exc [n] is typically much
smaller than the other known terms Ts [n], EHartree [n] and V̂en ,
with the hope that a reasonable simple guess for Exc [n] would
lead to useful results.
The construction of good exchange-correlation functionals represents nowadays an intensive field of the modern scientific research, but a certain number of exact properties have been well
established, and must be used as guidelines in this hard work.
Among the known properties of the exchange-correlation functional are the coordinate scaling conditions first obtained by Levy
and Perdew [17]
Ex [nλ ] = λEx [n]
,
Ec [nλ ] < λEc [n]
for λ < 1
Ec [nλ ] > λEc [n]
for λ > 1
(1.20)
14
1.5 EXACT PROPERTIES FOR Exc [n]
where nλ = λ3 n(λr) is a scaled density normalized to the total
number of electrons. Another important property of the exact
functional is represented by the one-electron limit
Ec [n(1) ] = 0
,
Ex [n(1) ] = −EHartree [n(1) ]
(1.21)
n(1)
where
is the one-electron density. These two latter conditions ensure that there is no self-interaction of one electron with
itself, and are satisfied within the Hartree-Fock approximation,
but not by standard local density and gradient corrected functionals. It also exist a lower bound which the exchange-correlation
functional must satisfied [18, 19]:
Z
2
Ex [n] > Exc ][n] > −1.68e dr n(r)4/3 ;
(1.22)
LDA and many (but not all) GGAs satisfy 1.22.
An important feature of the Exc [n] functional which all local or
semilocal approximations failed to reproduce is the discontinuity
of the functional derivate with respect to the electronic density
[20–22]:
δExc [n]
δExc [n]
+
−
−
= V̂xc
− V̂xc
= ∆xc (1.23)
δn
δn
N+δ
N−δ
where δ is a small positive shift in the number of electrons. Since
the one-electron kinetic energy functional has a similar discontinuity in the same functional derivative when crossing an integer
number of electrons, it is possible to write that
δE[n]
δE[n]
∆=
−
= ∆KS + ∆xc ,
(1.24)
δn N+δ
δn N−δ
where ∆KS refers to the kinetic energy discontinuity. ∆ defined
in 1.24 represents the true band gap in a solid; nevertheless ∆xc is
by construction lacking in any current approximated functional,
be it LDA, gradient-corrected or some other type. It is reasonable
to think that this missing term is responsible, for a large part, of
the band gap problem, at least in common semiconductors and
insulators.
All these properties serve as constraints and guidelines for the
construction of new approximations. Furthermore many other
properties are known, and Readers which are interested can find
more detailed informations in ref. [23].
15
1.6 LOCAL SPIN DENSITY APPROXIMATION
1.6
LOCAL SPIN DENSITY APPROXIMATION
The Hohenberg and Kohn and Sham theory [6, 9] was developed only in the spinless limit, and in cases where magnetic effects due to the presence of atoms with non-zero spin moments
become important, an extension of the theory is necessary. Great
work has been done in order to reformulate density functional
theory in the local density approximation for spin dependent systems [24, 25]; such extension is known as Local Spin Density (LSD)
approximation.
Traditional DFT, as we have seen, is based on two fundamental
theorems, namely that the ground state wavefunction is a unique
functional of the electronic charge density, and that there exists a
ground state energy functional which is stationary with respect
to variations in the charge density. These results can be generalized to the spin dependent case by replacing the scalar effective
one-electron potential V̂eff (r) in equation 1.8 by the spin dependent effective single-particle potential
Z
n(r 0 )
eff
2
V̂σ (r) = V̂en (r) + e dr 0
+ V̂σxc (r) ,
(1.25)
|r − r 0 |
where the charge density n(r) is intended as the sum of spin
densities n↑ (r) and n↑ (r), with
nσ (r) =
occ
X
|Ψi,σ (r)|2
,
(1.26)
i,σ
and Ψi,σ (r) spin dependent Kohn-Sham one electron orbitals. The
sum is over all occupied orbitals with spin σ. To obtain a reasonable approximation for the spin dependent potential V̂σeff (r) we
take the external potential V̂en (r) as slowly varying and divide
the electronic system into small boxes. Within the box centered
in r, the electrons can be considered to form a spin polarized homogeneous electron gas of local density n(r), and if εxc (n↑ , n↓ )
is the exchange-correlation energy per particle of such a system,
the spin dependent exchange-correlation potential is given by
V̂σxc (r) =
d
[n(r)εxc (n↑ , n↓ )]
dnσ (r)
,
(1.27)
16
1.6 LOCAL SPIN DENSITY APPROXIMATION
so that the spin dependent exchange-correlation energy functional
of the system can be written in a similar way as equation 1.14. In
terms of the quantity ζ(r) = [n↑ (r) − n↑ (r)]/n(r) which is proportional to the degree of polarization, the exchange-correlation
potential in 1.27 can be approximated by [26]
1 δ(rs )ζ(r)
0.611
β(rs ) ±
,
(1.28)
V̂σxc (r) = −
rs
3 1 ± 0.297ζ(r)
where ± refer to spin ↑ and ↓ respectively, and functions β(rs )
and δ(rs ) are parameterized in terms of the rs as follow:
11.4
β(rs ) = 1 + 0.0545rs ln 1 +
rs
rs
δ(rs ) = 1 − 0.036rs + 1.36
.
(1.29)
1 + 10rs
During the course of years, there have been other parameterized forms proposed for V̂σxc (r), but the uncertainties introduced
by the different choices remain smaller than the ones generated
by the LSD approximation itself. In the same manner one can
think to generalized gradient corrected or other type of exchangecorrelation functionals for spin dependent systems.
Returning for a moment to the previous section regarding exact properties, for a spin-dependent system, the exact exchange
energy obeys the spin-scaling relationship [27]
Ex [n↑ , n↓ ] =
Ex [2n↑ ] + Ex [2n↓ ]
2
,
and the same Lieb-Oxford bound 1.22 holds.
(1.30)
17
2
D F T C A LC U L AT I O N S
Once Density Functional Theory has been established, the next
problem is how to make its implementation as simple and less
computationally expensive as possible. A great number of technicalities have been studied to make DFT one of the most efficient and widely used theoretical instruments for ab-initio investigations in the field of condensed-matter and modern materials
design.
In this second chapter, we present some technical aspects concerning practical DFT calculations, e.g. how to improve the exchange correlation functionals seen so far and compute electronic
structures, how to choose proper basis functions for expanding
one-electron Kohn-Sham orbitals up to optimization procedures.
We start by presenting the Perdew Burke Ernzerhof exchange correlation functional [28], better known as PBE, which probably
represents nowadays the most popular and reliable GGA implementation. Trying then to overcome local and semilocal approximations, hybrid functionals provide a mix of local density and
exact non-local Hartree-Fock exchange; among them, we will focus our attention on Adamo Barone PBE0, Heyd Scuseria Ernzerhof HSE and Lee Yang Parr B3LYP exchange-correlation density
functionals. We then continue describing various methods used
to treat Coulomb interaction between ions and electrons, such as
the Pseudopotentials (PP) and the Projector Augmented Waves
(PAW). Methods to calculate ground state electronic structure
(such as self-consistency, diagonalization of Hamiltonian and direct minimization) are then illustrated, so that the problem of
optimization of atomic positions can be tackled. The Berry phase
formalism and the related modern theory of ferroelectric polarization and Born Effective Charge tensor are dealt with in the last
sections of this chapter.
All calculations in this work have been performed using the
Vienna Ab-initio Simulation Package (VASP), so some informations
18
2.1 PBE FUNCTIONAL
about this code are reported in the final paragraph, with a list of
references for further readings.
2.1
PBE FUNCTIONAL
In the Kohn-Sham density functional theory, as mentioned several times, only the exchange-correlation energy Exc = Ex + Ec
as functional of the electron spin densities n↑ (r) and n↓ (r) must
be approximated. A gradient corrected functional for a spindependent system is generally written in the form
Z
GGA
Exc [n↑ , n↓ ] = dr f(n↑ , n↓ , ∇n↑ , ∇n↓ ) ,
(2.1)
as seen for equation 1.19 in the spinless limit. Compared to local
spin density (LSD) approximation, GGA’s usually improves total
energies, cohesive energies, energy barriers and structural energy
differences [16, 29–31] correcting bond strengths and lengths [32]
with respect to simple local density based functionals (see discussion regarding 1.19). However, cases in which GGA’s overcorrect
LSD predictions could occur [33].
To facilitate practical calculations, the functional f in equation
2.1 must be parameterized through analytic functions of n(r) (or
equivalently of rs ) as for εxc . Despite parameterizations for the
latter are well established (see for example 1.15), the best choice
for the functional f(n↑ , n↓ , ∇n↑ , ∇n↓ ) is still a matter of debate.
A first-principles GGA can be constructed by starting from the
following second-order density-gradient expansion (GEA) for the
exchange-correlation hole surrounding each electron in a system
of slowly varying density [34]
XZ
0
LSD
[n
,
n
]
=
E
[n
,
n
]
+
dr Cσ,σ
EGEA
↑
↓
↑
↓
xc
xc
xc (n↑ , n↓ )
σ,σ 0
2/3 2/3
×∇nσ ∇nσ 0 /nσ nσ 0
,
(2.2)
and then cutting off its spurious long-range parts to satisfy sum
rules on the exact hole. Langreth and
R Perdew [35] showed that
the GEA hole violates the sum rule du nc (r, r + u) = 0 because
of its spurious long-u behavior, as well pointed out in Figure 3,
where, moreover, the cutoff in the GGA is also evident for z ∼ 10.
19
2.1 PBE FUNCTIONAL
Figure 3: Spherically averaged exchange hole density nx for s =
|∇n|/2kF n = 1 (s is the reduced density-gradient) in LSD (circles), GEA (crosses) and GGA (solid line) as function of the
reduced electron-electron separation on the scale of the Fermi
wavelength z = 2kF u. (From [34]).
One of the first GGA parametrization derived with this procedure was the Perdew-Wang 1991 (PW91) [36], but it presents
some problems: (1) the derivation is very long, (2) the analitic
function f in 2.1, fitted through the numerical results of the realspace cutoff, is complicated and nontransparent, (3) f is overparameterized, (4) although the numerical GGA correlation energy functional behaves properly under Levy’s uniform scaling
to the hight-density limit (see 1.20), its analytic parameterization
(PW91) does not, (5) it describes the linear response of the uniform electron gas [37] under small density’s variations less satisfactorily than LSD (does). PW91 functional was designed to
satisfy as many exact conditions as possible, but the semilocal
form of equation 2.1 is too restrictive to reproduce all the known
properties of the exact functional [34], so improvements can be
20
2.1 PBE FUNCTIONAL
carried on satisfying only those conditions which are energetically significant. This guideline led Perdew, Burke and Ernzerhof to propose the so called PBE exchange-correlation functional
[28], which generally solves all PW91’s problems, is simpler and
numerically very close to it.
PBE correlation term is usually written in the form
Z
PBE
Ec [n↑ , n↓ ] = dr n[εunif
(rs , ζ) + H(rs , ζ, t)] ,
(2.3)
c
where εunif
is the correlation functional for the homogeneous
c
electron gas, rs is the known local Seitz radius (n = 3/4πr3s =
k3F /3π2 ), ζ = (n↑ − n↓ )/n is the relative spin polarization, and
t = |∇n|/2φks n is a dimensionless density gradient. Here φ(ζ)
q =
4kF
π
h2 /me2 ).
[(1 + ζ)2/3 + (1 − ζ)2/3 ]/2 is a spin-scaling factor, and ks =
is the Thomas-Fermi screening wave number (a0 =
The term H in 2.3 can be written as
e2
β 2
1 + At2
3
,
H =
γφ ln 1 + t
a0
γ
1 + At2 + A2 t4
(2.4)
where
A=
β
[exp {−εunif
/(γφ3 e2 /a0 )} − 1]−1
c
γ
(2.5)
and γ ≈ 0.031091 β ≈ 0.066725. Ansatz 2.4 is formulated so that
the following known exact limits are verified: (a) in the slowly
varying t → 0 limit H is given by its second-order gradient expansion H → (e2 /a0 )βφ3 t2 ; (b) in the opposite t → ∞ limit H →
−εunif
, making correlation vanish; (c) under uniform scaling to
c
the high-density limit [n(r) → λ3 n(λr) and λ → ∞] the correlation energy must scale to a constant, thus H → (e2 /a0 )γφ3 ln t2
in order to cancel the logarithmic singularity of εunif
.
c
The exchange term in PBE functional is constructed satisfying
four further conditions: (d) under the same uniform density scaling of condition (c), Ex must scale linearly as function of λ (see
the first relation in 1.20), so that
Z
EPBE
[n
,
n
]
=
dr nεunif
(n)Fx (s) ,
(2.6)
↑ ↓
x
x
where s = |∇n|/2kF n is another dimensionless density gradient,
and εunif
given by 1.16; (e) the exact exchange energy must
x
21
2.1 PBE FUNCTIONAL
obey the spin-scaling relationship 1.30; (f) LSD linear response
of the spin-unpolarized uniform electron gas for small density
variations around the uniform density requires that as s → 0
Fx (s) → 1 + µs2 with µ ≈ 0.21951; (g) the Lied-Oxford bound
1.22 is verified only if Fx (s) 6 1.804. A simple enhancement exchange factor Fx (s) which satisfies all these conditions is
Fx (s) = 1 + κ − κ/(1 + µs2 /κ)
,
(2.7)
where κ = 0.804.
The general form for the PBE exchange-correlation functional
is usually written as
Figure 4: Enhancement exchange-correlation factor showing GGA (PBE)
nonlocality (e.g. s dependence). Solid curves are referred to
PBE functional, while open circles denote the PW91. (From
[28]).
22
2.2 HYBRID FUNCTIONALS
Z
EPBE
[n
,
n
]
=
dr nεunif
(n)Fxc (rs , ζ, s)
↑
↓
xc
x
,
(2.8)
where Fxc (rs , ζ, s) defines the enhancement exchange-correlation
factor [16, 29]. LSD can be seen as a further approximation, replacing Fxc (rs , ζ, s) in 2.8 with its zero-gradient value Fxc (rs , ζ, 0).
The enhancement factor’s behavior is reported in Figure 4 for
ζ = 0 and ζ = 1 in the range of interest for real system (0 6 s 6 3
and 0 6 rs 6 10) where is compared with PW91 results, demonstrating their numerical similarity.
In our work, all calculations involving generalized gradient approximations have been performed using the PBE exchange correlation functional, which is also the starting point for hybrid
functionals such as PBE0 and HSE to go beyond local and semilocal density approximations, as we will see in the next section.
2.2
HYBRID FUNCTIONALS
In the phase diagram of the homogeneous electron gas [10],
the correlation contribution is stronger than or comparable to exchange energy only in the low-density limit (rs >> 0). This observation suggests that ab-initio calculations are more reliable as the
exchange term is better treated. Kohn-Sham density functional
theory, in its more simple implementations, typically uses local
or semilocal approximations for the exchange-correlation functional Exc [n↑ , n↓ ] of the electron spin densities, even though it
also provides one-electron orbitals from which a Fock integral or
"exact" exchange energy can be constructed. In general, given any
pair of spin densities n↑ (r) and n↓ (r), there is usually a unique
Slater determinant Ψ0 of one-electron Kohn-Sham orbitals which
yields those densities and minimize the expectation values of the
kinetic energy operator T̂ and of the exact Kohn-Sham exchange
energy
Z
n(r)n(r 0 )
e2
Ex = hΨ0 |V̂ee |Ψ0 i −
drdr 0
,
(2.9)
2
|r − r 0 |
where V̂ee is the electron-electron repulsion operator and n =
n↑ + n↓ . Hybrid functionals which incorporate some of this ex-
23
2.2 HYBRID FUNCTIONALS
act exchange provide a simple and accurate description of the cohesive energies, bond lengths, and vibration frequencies of most
molecules [38, 39]. The growing use of hybrids in quantum chemistry calculations demands a simple rationale to establish how
much exact exchange should be included. Becke [40] showed
that the proper starting point for hybrid theory is the adiabatic
connection formula
Z1
Exc =
dλExc,λ ,
(2.10)
0
where
Exc,λ = hΨλ |V̂ee |Ψλ i −
e2
2
Z
drdr 0
n(r)n(r 0 )
|r − r 0 |
,
(2.11)
connects the noninteracting Kohn-Sham system (for λ = 0 there
is no correlation) to the fully interacting real system (λ = 1)
through a continuum of partially interacting systems, all sharing a common electron density n(r). From 2.9 it is easily verified that Exc,λ=0 = Ex . Assuming that at the end-point (λ = 1)
Exc,λ=1 = EDFT
xc , the most simple hybrid functional which approximates equation 2.10 is
Ehyb
xc =
1
(Ex + EDFT
xc )
2
,
(2.12)
where for EDFT
it is possible to assume every local or semiloxc
cal exchange-correlation approximation. An adiabatic connection
formula as 2.10 takes into account that local or semilocal functionals are more accurate at λ = 1 where the exchange-correlation
hole is deeper and thus more localized around its electron than
at λ = 0 where the exchange nonlocality is dominant.
Every density functional approximation EDFT
has a couplingxc
constant decomposition kernel EDFT
like
equation
2.10, which
xc,λ
DFT
DFT
DFT
DFT
permits to define Ex
= Exc,λ=0 and Ec
= Exc − EDFT
.
x
Perdew Ernzerhof and Burke [41] proposed the following simple
model for the hybrid coupling-constant dependence:
DFT
DFT
Ehyb
)(1 − λ)n−1
xc,λ (n) = Exc,λ + (Ex − Ex
,
(2.13)
24
2.2 HYBRID FUNCTIONALS
where n > 1 is an integer which controls how rapidly the correction to density functional approximation due to exact exchange
vanishes as λ → 1. Then it follows immediately that
Z1
1
DFT
DFT
Ehyb
=
) ,
(2.14)
dλEhyb
xc
xc,λ = Exc + n (Ex − Ex
0
which is a rationale for mixing exact exchange with density functional approximations. Perdew and co-workers have next shown
that the optimum value of the n coefficient can be fixed a priori
taking into account that the fourth-order perturbation theory is
sufficient to get accurate numerical results for molecular systems
[41], so n = 4. This leads to a family of adiabatic connection
hybrids with the same number of adjustable parameters as their
density functional constituents (usually GGA’s):
1
GGA
Ehyb
+ (Ex − EGGA
)
xc = Exc
x
4
.
(2.15)
The idea of Adamo and Barone was to use PBE as GGA exchange correlation functional in 2.15, because all its parameters
(other than those in its local spin density LSD component) are
fundamental constants, as we have seen in the previous section.
In this way, Adamo and Barone hybrid functional, known in the
literature as PBE0, does not contain any adjustable parameter,
and probably is one of the most reliable functional currently available in the study of molecular and solid state structures along the
whole periodic table.
However, in large molecules and solids, the calculation of the
exact Hartree-Fock exchange is computationally very expensive,
especially for systems with metallic characteristics. Much work
has been done to overcome this drawback; one possible solution,
by Heyd Scuseria and Ernzerhof [42], is to develop a hybrid functional based on a screened Coulomb potential for the exchange interaction. In general, long-range Coulomb interactions can be calculated efficiently for extended systems using techniques based
on the fast multipole method (FMM), but unfortunately, this approach cannot be used for the Hartree-Fock exchange interaction.
Furthermore, HF calculations in metals suffer from a divergence
in the derivative of the orbital energies with respect to the wavevector k due to the divergence of the Fourier transform 4π/k2
of the 1/r Coulomb potential for k = 0. This singularity can
25
2.2 HYBRID FUNCTIONALS
be avoided by using a screened Coulomb potential, which has a
shorter range that 1/r. Heyd and co-workers, in their HSE functional, apply a screened Coulomb potential only to the exchange
interaction in order to screen the long-range contribution of the
HF exchange, leaving unscreened all the other terms, such as
Coulomb repulsion between electrons. The starting point is to
split the Coulomb operator into short- (SR) and long- (LR) range
components; a possible way is
1
erfc(ωr) erf(ωr)
= SR + LR =
+
r
r
r
,
(2.16)
where erfc(ωr) = 1 − erf(ωr) and ω is an adjustable parameter.
HSE hybrid functional performs the exact exchange mixing of
equation 2.15 only for short-range interactions in both HF and
DFT, starting from the PBE0 model. We rewrite 2.15 for PBE0 as
EPBE0
= aEx + (1 − a)EPBE
+ EPBE
xc
x
c
,
(2.17)
where the exchange term is given by EPBE0
= aEx + (1 − a)EPBE
,
x
x
and a = 1/4; the splitting into short- and long-range components
leads to
EPBE0
x
LR
PBE,SR
= aESR
(ω)
x (ω) + aEx (ω) + (1 − a)Ex
+EPBE,LR
(ω) − aExPBE,LR (ω)
x
.
(2.18)
Numerical tests based on realistic ω values (for example ω =
0.15) indicate that the long-range exchange contributions aELR
x (ω)
and aEPBE,LR
(ω)
of
this
functional
are
rather
small
and
tend
to
x
cancel each other. Neglecting these two terms in 2.18, the HSE
hybrid functional is written as
EHSE
xc
PBE,SR
(ω)
= aESR
x (ω) + (1 − a)Ex
(ω) + EPBE
(ω)
+EPBE,LR
x
c
.
(2.19)
Other rationales for mixing exact exchange with density functional approximations have been studied and proposed, sometimes with a little of empiricism. One of the most popular hybrid
functionals owing this class, is the Becke three-parameter hybrid
EB3
xc
LSD
= ELSD
) + ax (EGGA
− ELSD
)
xc + a0 (Ex − Ex
x
x
+ac (EGGA
− ELSD
)
c
c
,
(2.20)
26
2.3 THE ELECTRONIC GROUND STATE
usually known as B3, where the parameters a0 = 0.20 ax = 0.72
and ac = 0.81 were determined by fitting to a data set of measured cohesive energies. If the GGA components in 2.20 are chosen to be those of Lee Yang Parr exchange-correlation functional
LYP [43], the resulting hybrid functional, widely used in quantum chemistry calculations, is known as B3LYP.
In our study about organic ferroelectrics, calculations beyond
local and semilocal approximations have been performed through
all these hybrid functionals, and wherever possible, comparisons
with experimental results will be reported to estimate the reliability of these approximated models. Nevertheless, noncovalent
interactions as van der Waals forces play an important role in the
studied organics. These weak vdW interactions are a quantummechanical phenomenon with charge fluctuations in one part of
the system that are correlated with charge fluctuations in another.
The vdW forces at one point depend on charge events in another
region, and thus they are pure non-local correlation effects. The
exact density functional density contains the vdW forces; unfortunately we don’t have access to it, but only to its approximated parameterizations as LDA and GGA’s which, depending on the density in local and semilocal ways respectively, give no account of
the fully nonlocal vdW interaction. The first-principles approach
to treat vdW’s in DFT is the inclusion of a full non-local density
functional long-range correlation energy Enl
c [n] of the form
Z
0
0
0
(2.21)
Enl
c [n] = drdr n(r)φ(r, r )n(r ) ,
as implemented in the vdW-DF hybrid functional [44, 45]. The
kernel φ is given as a function of |r − r 0 |f(r) and |r − r 0 |f(r 0 ), with
f(r 0 ) function of the local density n(r) and of its gradient.
We benchmarked our simulations with an empirical technique
to account for van der Waals interactions in density functional
theory called Grimme’s corrections [46] and implemented in VASP
by T. Bucko et al. [47]. We will refer to it as vdW-G functional.
2.3
THE ELECTRONIC GROUND STATE
Keeping fixed the atomic positions, there are at least two different ways to find the electronic ground state. The first is to solve
27
2.3 THE ELECTRONIC GROUND STATE
self-consistently the Kohn-Sham equations 1.8, iterating on the
charge density n(r) (or equally the potential) until self-consistency
is achieved. The second is to directly minimize the energy functional with respect to the coefficients of the Kohn-Sham orbitals’
expansion (either plane waves or other proper basis sets) under
the constraint of orthogonality.
In the former case, supplying an initial charge density nin (r)
to the Kohn-Sham equations 1.8, an unique operator  is defined
such that
nout (r) = Â[nin (r)]
;
(2.22)
at self-consistency, n(r) = Â[n(r)] must hold. In this way, the simplest algorithm implies the use of nout as the new initial guess
for the charge density, e.g.
(i+1)
nin
(i)
= nout
,
(2.23)
where the superscripts refer to the iteration number. Unfortunately, there is no guarantee that the scheme 2.23 works properly,
and in general it does not. The reason is that algorithms for selfconsistency work only if the error in output δnout is smaller than
the error in input δnin . For the scheme in 2.23 the propagation
of output uncertainties is given by a relation as
δnout = Jδnin
,
(2.24)
and depending on the size of the largest eigenvalue eJ of the
matrix J with respect to the unity, the algorithm could converge
or not. Usually eJ > 1, and improved schemes must be taken into
account. The so called simple mixing generally works, although
sometimes slowly; it is based on the scheme
(i+1)
nin
(i)
(i)
= (1 − α)nin + αnout
,
(2.25)
where the value of α must be chosen empirically to get fast convergence; it is easily seen that the iteration converges if α < |1/eJ |.
Better results could be obtained with more sophisticated algorithms which use informations coming from many preceding iterations; among them, one of the most widely implemented is
the Direct Iteration in Inverse Space (DIIS) method [48].
Furthermore, when the wavefunctions, usually the PS wavefunctions, are expanded on a finite basis set (for example plane
28
2.4 OPTIMIZATION OF ATOMIC POSITIONS
waves), the Kohn-Sham equations 1.8 take the form of a secular
equation:
X
H(k + G, k + G 0 )Ψk,i (G 0 ) = εk,i Ψk,i (G 0 ) ,
(2.26)
G0
where H(k + G, k + G 0 ) are the matrix elements of the Hamiltonian operator. In this way, the problem of finding the electronic
ground state is reduced to the calculation of the lowest eigenvalues and eigenvectors of a Npw × Npw Hermitian matrix, where
Npw is the number of plane waves used in the expansion. This
task can be performed through the bisection-tridiagonalization
algorithms, which are implemented in many public-domain computer packages like LAPACK libraries. However, the CPU time
required to diagonalize a Npw × Npw matrix grows as N3pw , and
the storing requires a computer memory which scales as N2pw .
As a consequence, a calculation with more than a few hundred
plane waves becomes exceedingly time- and memory- consuming. For these reasons direct minimization methods have been
studied. The energy functional can be written as a function of
the coefficients in the basis set of the Kohn-Sham orbitals, and
directly minimized under the orthonormality constraints. In this
way, the problem is to find the minimum of
E0
= E(Ψk,i (G)) −
"
#
X
X
∗
λij
Ψk,i (G)Ψk,j (G) − δij
ij
,
(2.27)
G
with respect to the variables Ψk,i (G) and the Lagrange multipliers λij . In general, the gradient of E 0 is also available, so that specialized algorithms, such as steepest descendent conjugate gradient methods, can be used.
2.4
OPTIMIZATION OF ATOMIC POSITIONS
In all the previous discussion, we always assumed that ionic
positions are kept fixed, so that self-consistency leads to the correct electronic ground-state for that given structure. Usually the
atomic configurations are known from experimental studies, such
29
2.4 OPTIMIZATION OF ATOMIC POSITIONS
as X-ray scattering or neutron diffraction, and often one of the
first computational problems is to find the minimum of total energy as a function of atomic positions. Two considerations can
now be made. The first is that, if the starting structure belongs
to a given space group with given symmetry properties, forces
acting on atoms can never break such symmetry. The second is
that algorithms based on forces bring the system to the closer local minimum (the closer zero-gradient point) rather than to the
absolute minimum energy.
Given the Kohn-Sham total electronic energy functional E[n] as
in equation 1.7, the total energy of the system is
Etot [n] = E[n] + Enn
,
(2.28)
where the term Enn refers to the Coulomb ion-ion repulsion energy. We can write the force acting on the atom in the position
Ri as
Z
Fi = −∇Ri Etot = − dr n(r)∇Ri Ven (r)
−∇Ri Enn − F̃i
,
(2.29)
where the first and the last terms come respectively from explicit
and implicit derivation of the electronic energy functional E[n].
Implicit derivation is related to the implicit atomic positions dependence of Kohn-Sham one-electron orbitals; more explicitly,
one can see that [49]
XZ
F̃i =
dr [∇Ri Ψ∗k (r)(ĤKS − εk )Ψk (r)
k
+∇Ri Ψk (r)(ĤKS − εk )Ψ∗k (r)]
,
(2.30)
which clearly vanishes if Ψk (r) and εk are the ground-state eigenfunctions and eigenvalues of the Kohn-Sham Hamiltonian ĤKS .
In this way, only the expectation value of the Coulomb electronion interaction gradient ∇Ri Ven (r) and the Coulomb ion-ion repulsion energy gradient ∇Ri Enn give rise to the force acting on
the i-th atom.
Unfortunately, the term F̃i in 2.30 vanishes only if the groundstate charge density n(r) and wavefunctions Ψk (r) are perfectly
converged, and this is never the case in real calculations because
30
2.5 THE MODERN THEORY OF POLARIZATION
wavefunctions are expanded on a finite size basis set which cannot be complete. Nonzero F̃i terms are known as Pulay forces.
Nevertheless, it’s possible to see that Pulay contributions are identically zero if Kohn-Sham orbitals are expanded on a plane wave
basis set, because it doesn’t explicitly depend on the atomic positions Ri . Spurious contributions must be however taken into
account in practical calculations with localized basis sets.
2.5
THE MODERN THEORY OF POLARIZATION
The macroscopic electric polarization of materials plays a fundamental role in the phenomenological description of dielectrics.
The progress of the methods of electronic structure calculations,
and in particular of the density functional theory, have made possible accurate first-principles investigations of the ground-state
properties of interacting electron-nuclear systems. In this section,
we will briefly discuss some aspects of the quantum theory of
polarization of crystalline solids and the role assumed in this theory by the geometric Berry phase [50]; for more details on the
formalism based on the Berry phase, and for workable microscopic calculations of polarization changes in ferroelectric and
piezoelectric crystals we refer to the original works of King-Smith
and Vanderbilt [51] and of Resta [52].
Consider a crystal of volume V = NΩ, formed by an arbitrary
large number N of identical unit cells of volume Ω. The average
electric polarization of the crystal, e.g. the electric dipole per unit
volume, is related to the electronic charge density n(r) by the
expression


Z
1 X
P = Pion + Pel =
zj eRj − e
dr n(r)r , (2.31)
NΩ
NΩ
j
where e is the absolute value of the electronic charge and Rj are
the positions within the crystal of the nuclei of charge zj e. The
average polarization 2.31 is also called macroscopic polarization,
or simply polarization, of the crystal. The polarization vector P
as defined in 2.31 depends on the details of the unit cell chosen
to build up the crystal, whereas infinitesimal changes of polarization are independent of how the crystal has been assembled and
31
2.5 THE MODERN THEORY OF POLARIZATION
are thus bulk properties. For these reasons, all the physical effects
related to changes of polarization can be evaluated unambiguously
and compared with experimental measurements. What could
be measured experimentally are changes of polarization from a
centric crystal structure with symmetry inversion properties and
a polar one, in which that symmetry has been broken by polar
distortions. Let λ be a continuous parameter varying from 0 to
1 that denote the relative distortion between the two structures.
For any assigned value of λ, let Ψn (k, r, λ) indicate the KohnSham one-electron orbitals, and we will now focus on the change
of electronic polarization as λ varies. From the knowledge of
the parameter-dependent orbitals Ψn (k, r, λ), we can express the
electronic contribution to the average crystal polarization in the
form
Z
e
Pel (λ) = −
dr n(r)r
NΩ NΩ
2e X
= −
hΨn (k, r, λ)|r|Ψn (k, r, λ)i ,
(2.32)
NΩ
nk
where the factor 2 takes into account the spin degeneracy and the
sum is over all occupied bands of the semiconductor or insulator
under study. Indicating with un (k, r, λ) the periodic part of the
Bloch functions, equation 2.32 can be written as
2e X
Pel (λ) = −
hun (k, r, λ)|r|un (k, r, λ)i .
(2.33)
NΩ
nk
As we noted before, only changes in polarization have real physical meaning, so we are interested in variations of Pel (λ) with
respect to λ:
∂Pel (λ)
2e X
∂
=−
2Rehun (k, r, λ)|r| un (k, r, λ)i (2.34)
∂λ
NΩ
∂λ
nk
where Re stands for the real part. After some manipulations,
equation 2.33 can be recast in the usual form
∂Pel
2e X
∂
=−
2Imh∇k un (k, r, λ)| un (k, r, λ)i ; (2.35)
∂λ
NΩ
∂λ
nk
the total change ∆Pel in polarization is obtained by integrating
2.34 in dλ within the range 0 6 λ 6 1 and in the Brillouin zone.
32
2.6 THE VASP CODE
In practice, integration over the three-dimensional Brillouin zone
is carried out performing integrations over one variable, say for
example kz , once a number of special points are chosen for the
other two variables kx and ky . From these assumptions, the final
form that one can get for the n-th band contribution, after a little
of algebra, is
∆Pel
=
=
e
γn (C)
(2.36)
πab
I
e
Im hun (kz , r, λ)|∇kz ,λ un (kz , r, λ)i · dl
−
πab
C
−
where γn (C) is the Berry phase of the cell-periodic wavefunctions
moving along the circuit C identified by the rectangle −π/c 6
kz 6 π/c and 0 6 λ 6 1 in the (kz , λ) space. From a physicalmathematical point of view, the Berry phase is the phase acquired
by a quantum system described by a parameter-dependent Hamiltonian moving along a circuit C on a given adiabatic parameterdependent surface.
A quantity strongly related to the macroscopic polarization is
the Born Effective Charge tensor defined as
Z∗i,αβ =
Ω ∂Pα
|e| ∂ui,β
,
(2.37)
e.g the ratio between the change in the α-th component of polarization due to an infinitesimal displacement u of the i-th atom
in the β direction. The knowledge of the Born Effective Charge
tensor could be useful in the determination of atoms which play
an active role in ferroelectric transitions, because for these atoms
the Z∗ charge is often much larger than the nominal one.
2.6
THE VASP CODE
The Vienna Ab-initio Simulation Package (VASP) is a package to
perform ab-initio quantum mechanical simulations using pseudopotentials or the projection augmented wave method and a
plane wave basis set. The approach implemented in VASP is
based on the local density approximation with the free energy as
variational quantity and an exact evaluation of the instantaneous
33
2.6 THE VASP CODE
electronic ground state. VASP uses efficient matrix diagonalization schemes and an efficient Pulay charge density mixing [48].
The interaction between ions and electrons is described by ultrasoft Vanderbilt pseudopotentials (US-PP) or by the projector augmented wave (PAW) method. Both these methods allow for a
considerable reduction of the number of plane waves per atom
for transition metals and first row elements. Forces and the full
stress tensor can be calculated by VASP and used to relax atoms
in their instantaneous ground-state.
This short description of the VASP is how G. Kresse M. Marsman and J. Furthmüller, the authors of the code, present it in
the official website [53]. The Reader interested in further readings about how specific algorithms and schemes have been implemented in VASP can find more informations in the original
works [54] and [55].
34
3
S Y M M E T R Y A N A LY S I S
After the work of Landau [56], the natural framework to deal
with displacive structural distortions is that of symmetry mode
analysis. Modes are collective correlated atomic displacements.
Structural distortions in every structure can be decomposed into
contributions coming from different modes associated to different symmetries given by the irreducible representations of the
parent space group. Furthermore, it is possible to distinguish
primary and secondary, or induced, distortions, which will in
general respond in different ways to external perturbations.
For the systems studied in this thesis, one can find a highsymmetry centric structure from which, through polar distortions, one obtains the ferroelectric phase. For each system we
determine the relevant polar distortions and their characterization in terms of different symmetry modes. Each mode is then
individually investigated and its contribution to the total polarization is calculated. Symmetry analysis are performed using
specific tools from the Bilbao Crystallographic Server [7].
3.1
THE PSEUDO TOOL
If a crystal structure characterized by a symmetry space group
H is such that its atomic positions Ri can be described as R0i + ui ,
where ui are small displacements and where the virtual structure
with atomic positions R0i belongs to a higher symmetry space
group G > H, then H is said to have pseudosymmetry G, or equivalently H is pseudosymmetric for the space group G.
The detection of pseudosymmetries can be useful for many
purposes. For example, the knowledge of a pseudosymmetry
can be a tool to predict possible structures and symmetries involved in transitions. Furthermore, it can also be a valid tool
to identify ferroic materials such as ferroelectrics and ferroelas-
35
3.2 THE AMPLIMODES TOOL
tics, and to determine optimized virtual parent structures. The
Bilbao Crystallographic Server [7] provides a computer software
for pseudosymmetry search in a given structure, the so-called
PSEUDO tool [57].
PSEUDO aims to find a pseudosymmetry of a given distorted
structure and to find a virtual parent high-symmetry structure.
The software works properly and correctly detects pseudosymmetries if the maximal atomic displacement ui - that relates the
input structure and the high-symmetry one - is usually not larger
than 1 Å, although it is possible to set up a larger threshold. In
the latter case, a check on the conservation of the chemical connectivity is necessary to avoid non-physical configurations. In
principle, the program is not suited to investigate pseudosymmetry in structures with order-disorder distortions, although some
tricks, such as the averaging of atomic positions, can be used.
If G is a possible pseudosymmetry of the given low-symmetry
space group H, a so-called left coset decomposition [58] of G with
respect to H exists, and a set {1, g2 , · · · , gn } of coset representatives can be chosen so that
G = H + g 2 H + · · · + gn H .
(3.1)
The set {1, g2 , · · · , gn } contains all the operations of G which are
not symmetry operations for H.
Denoting S as the input structure, the transformed structures
gi S are calculated by the PSEUDO program and compared with
the original structure S. If the displacements between S and the
structures gi S are below a given tolerance, then the space group
G is considered as pseudosymmetric. Given an input structure
with space group H, several pseudosymmetries G below a predefined tolerance exist, but the one with minimal displacements
has to be chosen. Usually one refers to the space group H as
subgroup, whereas to G as supergroup.
3.2
THE AMPLIMODES TOOL
Once a reference paraelectric phase has been found by PSEUDO,
a full characterization in terms of symmetry-adapted polar modes
can be performed using the AMPLIMODES tool [59] in the Bilbao
Crystallographic Server [7].
36
3.2 THE AMPLIMODES TOOL
AMPLIMODES performes the symmetry modes analysis of
any distorted structure derived by a displacive type structural
transition. The analysis consists in the decomposition of the
symmetry-breaking distortion into contributions from different
polar modes. Starting from the high- and low- symmetry structures, AMPLIMODES determines the atomic displacements ui
that relate the two structures, defines a proper basis of symmetryadapted modes for expanding the displacement field, and calculates amplitudes and directions of the polarization vectors.
In Appendix C, we report an example on how the PSEUDO
and AMPLIMODES tools work in the case of the CBDC, an organic molecular crystal recently found to be ferroelectric [60].
37
Part II
Computational Results
and Analysis
38
4
P O L A R D I S T O R T I O N S I N H YDROGEN BONDED ORGANIC FERROELECTRICS
The property of ferroelectric polarization switchable by an applied electric field, e.g ferroelectricity, is the basis of a wide range
of device applications, including non-volatile computer memory,
ultrasonic imaging, nanomanipulation and optical devices [61].
The first discovery of a ferroelectric material goes back to 1920
with the sodium potassium tartrate tetrahydrate (NaKC4 H4 O6 ·
4H2 O), better known also as Rochelle salt [2, 62]. Despite the fact
that this is the first-discovered ferroelectric material, it is one of
the most complicated known to date, and research in this field
soon focused on simpler materials, such as phosphates and arsenates [63, 64]. A typical example is potassium dihydrogen phosphate, KH2 PO4 , also known as KDP [65], which contains hydrogen bonds and where different arrangements of hydrogens result
in different orientations of the dipolar units. After the discovery
of ferroelectricity in barium titanate, BaT iO3 , with a polarization
as large as 27 µC/cm2 in the tetragonal phase [66], researchers focused their attention on the new class of perovskite oxygen-based
ferroelectrics, which are by far the most investigated class of ferroelectric materials and the most important for current device applications. In the last few years, the growing interest in materials
design lead scientists to study ferroelectrics that are potentially
cheaper, more soluble, less toxic, lighter and more flexible, such
as organic ferroelectrics, which presently plays a leading role in
modern materials science.
When comparing with inorganic compounds, we note that organic materials have been synthesized in large numbers but ferroelectric properties have been found (or searched for) only rarely.
The other feature is their tendency to form highly anisotropic
structures with low lattice symmetry. Despite their occasional
crystallization in polar structures, their dielectric properties and
39
4.1 STRUCTURAL PROPERTIES
possible ferroelectricity were seldom examined, especially in the
case of hydrogen-bonded organic ferroelectrics. The aim of this
chapter is to shed light on possible microscopic mechanisms at
play in hydrogen-bonded organic ferroelectric substances from
a theoretical point of view. In closer detail, we investigate the
properties of four prototypical organic molecular crystals such
as 1-cyclobutene-1,2-dicarboxylic acid (CBDC, C6 H6 O4 ) [67], 2phenylmalondialde- hyde (PhMDA, C9 H8 O2 ) [68], 2-fluoro-1,3cyclohexadione (2-FCHD, FC6 H7 O2 ) [69] and 4,5-dihydroxycyclopentenetrione (croconic acid, C5 H2 O5 ). The latter was the first
discovered single-component organic ferroelectric exhibiting a
large spontaneous polarization (as large as 21 µC/cm2 [70, 71]).
In the next chapter its electronic properties will be investigated
beyond the local and semilocal approximations through a theoretical study based on hybrid functionals.
4.1
STRUCTURAL PROPERTIES
We recall that ferroelectricity requires both a polar crystal structure as well as the switchability of the electric polarization. This
latter condition automatically implies the existence of a sufficiently close (in term of atomic displacements) paraelectric state
as an intermediate structure along the +P → −P switching path.
In organic crystals, routes towards ferroelectricity were first suggested by Zikmund et al. and followed by Horiuchi, Tokura et
al..
The β-diketone enol O = C − C = C − OH moieties (where =
and − refer to double and single bonds respectively) could be
used as building-block for a certain number of hydrogen-bonded
organic molecular crystals. Similarly, it is possible to think of two
carboxylic groups bonded to the C = C bond as in O = C − C =
C − C − OH. The reason is that such moieties could form infinite
hydrogen-bonded molecular chains, and the collective site-to-site
transfer of hydrogen atoms from the OH · · · O bonds to O · · · HO
bonds switches the spontaneous polarization and triggers the ferroelectric ordering of the lattice. By applying this procedure to
single-component molecules containing these moieties, the latter
40
4.1 STRUCTURAL PROPERTIES
substances can be taken as candidates. Their structures in the
polar phase are reported in Figures 5-8.
The CBDC molecular unit is formed by a main planar fourmembered ring of carbon atoms, similar to the planar cyclobutene
molecule, and two carboxylic carbons with formula HO − C = O
(Figure 5). The molecular units are part of infinite chains of
molecules linked by intermolecular hydrogen bonds and related
by glide plane. It is important to note that inside each molecular
unit the two carboxylic groups are bound by another hydrogen
bond, hereafter labeled as intramolecular. Intermolecular and intramolecular OH · · · O bonds alternatively connect the carboxylic
groups along the crystallographic [2, 0, 1] direction, forming in
this way unidirectional linear chains. The C-centered monoclinic
crystal system owing to the Cc space group restricts the polar
axis within the ac plane, the spontaneous polarization being, by
symmetry, orthogonal to the b axis. Without the two hydroxyl
protons, e.g. the hydrogen atoms involved in hydrogen bonds,
the crystal can adopt the C2 symmetry so as to restore the inversion center. From the analysis with the PSEUDO tool we find
that the centric structure belongs to the space group C2/c, and
the inversion symmetry is located on the intermolecular OH · · · O
bonds. The structures in the polar and centric phases are related
by a maximum atomic displacement of 0.43 Å, and a total distortion amplitude of 1.65 Å.
As for PhMDA (Figure 6), its molecular unit is formed by a
planar phenyl group, six carbon atoms arranged in a planar ring,
each of which is bonded to one hydrogen atom. The phenyl
group is linked to a linear hydrogen bonded chain of β-diketone
enol moietes, as explained above; the hydrogen bonds link the
molecular units along the [102] and [102] crystallographic directions. Molecular chains of different orientations are related one
to the other by a twofold screw axis, and their net polarity is
directed along the c-direction in the orthorhombic crystal (space
group Pna21 ). As for CBDC and all other organic ferroelectrics
treated in this work, the molecule without the hydroxyl proton
can restore the C2 symmetry and the inversion symmetry on the
hydrogen-bonding center. We find that the structure in the polar
phase shows a Pbcn pseudosymmetry with a maximum atomic
displacement of 0.25 Å.
41
4.1 STRUCTURAL PROPERTIES
Figure 5: CBDC.
Figure 7: 2-FCHD
Figure 6: PhMDA
Figure 8: Croconic acid
42
4.1 STRUCTURAL PROPERTIES
Among the four crystals, 2-FCHD is probably the less investigated. Its molecular unit, shown in Figure 7, is formed by a nonplanar six-membered ring of carbon atoms. Three of these are
linked with two hydrogen atoms in a methylene group; among
the three remanent carbons, two forms with oxygen atoms the
carboxylic groups necessary for the hydrogen-bonded infinite
chains, and the third is involved in a carbon-fluorine bond. The
non-planarity of the carbon ring is substantially due to steric interactions between hydrogen atoms which tend to avoid each
other, so as to distort the structure. For the 2-FCHD, the symmetry of the Cc space group restricts the polarization to be along
the c-axis. When restoring a mirror plane, the structure shows a
C2/c pseudosymmetry, with a maximum atomic displacement of
0.97 Å and a total polar distortion amplitude of 2.91 Å.
Despite its discovery 180 years ago and its recent use in nearinfrared-absorbing dyes, the croconic acid was successfully crystallized only few years ago. To our knowledge, the croconic
acid was the first example of ferroelectricity achieved by proton
transfer in a single-component molecule. There are two crystallographically independent hydrogen bonds; one almost parallel
to the polar c-axis forming a linear hydrogen-bonded chain, and
the other zigzags along the c-axis forming coplanar molecular sequences. The combined hydrogen-bond network forms a zigzag
sheet. In the polar phase, the structure belongs to the Pca21
space group, and by symmetry, the spontaneous polarization is
directed along the c-axis. When hydroxyl protons are ignored, an
additional mirror plane can survive perpendicularly to the pentagon forming the molecular unit’s body, and the inversion center
can be restored in the structure with pseudosymmetry Pbcm.
An almost common character of these organic materials is the
distance O · · · O between head-to-tail oxygens that link the molecular units. In all cases, this distance ranges from 2.53 to 2.64 Å.
Thereby, we expect all these crystals to allow the hydroxyl proton to hop between the two potential minima corresponding to
opposite polarization states. In this way, each chain can host the
bistability necessary for ferroelectricity: all the hydrogen-bonds
switch from the OH · · · O to O · · · HO form at once. This proton
transfer (also called proton tautomerism) for polarity reversal involves one proton per molecule in PhMDA and 2-FCHD, whereas
it involves two protons per molecule in CBDC and croconic acid.
43
4.1 STRUCTURAL PROPERTIES
Figure 9: Schematic ball-and-stick model of two CBDC molecular units.
Thin dashed lines refer to hydrogen-bonds. Labels are consistent with Table 1. (The figure is adapted from [72])
For CBDC, we have also studied the structural properties from
a theoretical point of view within and beyond the local density
approximation, and we have compared our results with those
obtained from X-ray diffraction analysis by Horiuchi et al. [60] at
the temperature of 50 K.
In Table 1 we report bond lengths as obtained by ab-initio calculations using standard PBE for semilocal density approximation,
and HSE and van der Waals with Grimme’s correction VdW-G
hybrid functionals. The last column reports the experimental results at the temperature of 50 K. It is evident how differences
in bond lengths between theoretical and experimental data are
larger when hydrogens atoms are involved. This is due to the fact
that experimental determinations of atomic positions for lighter
ions are very difficult and sometimes greatly underestimated. On
the other hand, all C − C, C − O and O − O bonds are in good
agreement within few percents. In our calculations, all ionic positions were relaxed so as to minimize the systems’ total energies
and forces. Furthermore, for each functional, we calculated mean
absolute deviations (MADs) and root-mean square (RMS) errors
with respect to the experimental data set, finding an improvement in HSE calculations, despite of a higher computational cost.
A deeper study with hybrid functionals will be presented specifically for the croconic acid in the next chapter. Here, all our
44
4.1 STRUCTURAL PROPERTIES
theoretical results are obtained using the semilocal density approximation (PBE) for the exchange-correlation functional.
PBE
Bond lengths
C(5)-O(1)
C(5)-O(2)
C(1)-C(5)
C(1)-C(4)
C(1)-C(2)
O(1)-HO(1)
C(4)-H(41)
C(4)-H(42)
HO(1) · · · O(3)
O(1)O(3)
C(6)-O(4)
C(6)-O(3)
C(2)-C(6)
C(2)-C(3)
C(3)-C(4)
O(4)-HO(4)
C(3)-H(31)
C(3)-H(32)
HO(4) · · · O(2’)
O(4)O(2’)
Estimator
MAD
RMS
HSE
VdW-G
Expt(50 ◦ K)
1.32
1.25
1.47
1.51
1.36
1.05
1.10
1.10
1.49
2.54
1.32
1.25
1.47
1.51
1.57
1.05
1.10
1.10
1.53
2.58
1.32
1.23
1.47
1.51
1.35
0.85
0.95
0.97
1.76
2.60
1.31
1.23
1.47
1.51
1.57
0.78
0.98
0.99
1.84
2.61
◦
1.32
1.25
1.47
1.50
1.36
1.05
1.10
1.10
1.47
2.52
1.32
1.25
1.47
1.51
1.57
1.05
1.10
1.10
1.53
2.58
A
1.31
1.23
1.46
1.50
1.35
1.02
1.09
1.09
1.51
2.53
1.30
1.24
1.46
1.50
1.57
1.02
1.09
1.09
1.58
2.60
0.09
0.14
A
0.08
0.12
◦
0.09
0.13
Table 1: Comparison between theoretical and experimental results about
structural properties of CBDC molecular units. Mean absolute deviation (MAD) and root-mean square (RMS) error are
reported for each functional with respect to the experimental
set of data [60].
45
4.2 THE FERROELECTRIC POLARIZATION
4.2
THE FERROELECTRIC POLARIZATION
First of all, we report some experimental results about the spontaneous polarization for CBDC, PhMDA [60], and croconic acid
[71]. For 2-FCHD, to the best of our knowledge, our study represents the first attempt to describe and characterize the polarization in this compound (for CBDC and PhMDA it represents the
first theoretical work), so that experimental results for comparison are not available. For the other three compounds, despite
their distinct hydrogen-bonding configurations, the ferroelectric
nature has been directly addressed by the electric polarization P
versus electric field E hysteresis measurements at room temperature. The hysteresis loops for the three compounds well identify
a ferroelectric behavior, with remanent polarization Ps of 0.4 and
2.8 µC/cm2 along x- and z-axes for CBDC, 9.0 µC/cm2 along zaxis for PhMDA and 21 µC/cm2 along z-axis for croconic acid.
In single-component molecular crystals, it is not possible to
evaluate the spontaneous polarization simply by taking the total vector sum of dipole moments of molecular units in the unit
cell. Rather, as briefly discussed, the modern theory of polarization demands precise knowledge of the electronic structure in
the crystalline phase, and so it is necessary to evaluate the spontaneous polarization from first-principles calculations using the
Berry phase formalism. The Kohn-Sham equations were solved
using the projector augmented wave pseudopotentials and the
PBE generalized gradient density approximation to the exchangecorrelation potential. We used a plane-wave cut-off of 400 eV and
k-point meshes of (6, 2, 4) (4, 2, 6) (6, 3, 5) and (5, 3, 2) for CBDC,
PhMDA, 2-FCHD and croconic acid respectively. All atomic positions were optimized until the forces were below 0.01 eV/Å. Test
calculations were performed to estimate the effect of (1) electronic correlations beyond the local density approximation and
(2) van der Waals corrections by using HSE and VdW-G hybrid
functionals. In both cases the changes in the magnitude of polarization were found to be less than few percents, confirming the
basic physics explained below.
Let’s begin with CBDC and PhMDA. In order to gain insight
into their ferroelectricity, we compare the relaxed centric and polar structures shown in Figure 10(a-b) for CBDC and in Figure
10(c-d) for PhMDA. In a Landau-type structural phase transition
46
4.2 THE FERROELECTRIC POLARIZATION
Figure 10: Schematic ball-and-stick model of CBDC (top) and PhMDA (bottom). (a) and (c) refer to centric and (b) and
(d) to polar structures. Dashed guiding-eye lines refer
to the position of relevant hydrogens contributing to polarization; arrows in (b) e (d) indicate important polar
distortions.
47
4.2 THE FERROELECTRIC POLARIZATION
from a high-symmetry parent structure to a low-symmetry one, it
is useful to introduce a global distortion parameter λ, labeling the
centric phase with λ = 0 and the polar one with λ = 1. In CBDC
two types of hydrogen bonds are present, namely the intramolecular and intermolecular bonds as underlined above; in the polar
state, they shift toward the molecular units on the right, as shown
by the short arrows in Figure 10(b). There is another cooperative
atomic distortion, schematically reported in Figure 10(b) by the
curved arrow, hereafter referred to as molecular buckling. For PhMDA, the hydrogen sitting between two molecular units in the
polar structure shifts toward one of its neighboring units, as indicated by short arrows in Figure 10(d). Two other relevant atomic
distortions come into play, as shown by the curved arrows, both
Figure 11: Variation of total energy (top) and of polarization (bottom) as
a function of the amplitude of the polar distortion between
centric (λ = 0) and polar (λ = ±1) configurations.
48
4.2 THE FERROELECTRIC POLARIZATION
Figure 12: Same plots as in Figure 11 for the croconic acid (left) and
2-FCHD (right).
tending to deform the molecular units. One acts on the planar
phenyl group, while the other distorts the β-diketone enol moieties. For both compounds, then, we will demonstrate - on the
basis of a symmetry-mode analysis - that three different types of
distortion mainly contribute to the total ferroelectric polarization.
In Figure 11 we report the variation of the total energy from the
centric (λ = 0) to the polar (λ = ±1) structures as a function of
the amplitude of the polar mode. We calculated the electronic
structures for a large number of λ values that fall between the
paraelectric and the real ferroelectric distorted structure. For both
materials we find a bistable energy profile characteristic of a ferroelectric material, with an energy barrier of about 0.3 eV/unit cell,
suggesting that the polarization should be switchable upon the
application of a moderate external electric field. We here recall
49
4.2 THE FERROELECTRIC POLARIZATION
that experimental values for the coercive electric field range from
4 and 6 KV/cm. Regarding the polarization, for CBDC it is in
the ac plane with a magnitude of P = 14.3 µC/cm2 , while for
PhMDA it is along the c-axis and equal to 7.0 µC/cm2 ; from Figure 11 it is evident how polarization behaves almost linearly as a
function of the polar mode amplitude λ.
Figure 13: The same as Figure 10 for croconic acid (top) and 2FCHD (bottom).
We carried out a similar study for the other two compounds. In
Figure 12, energy barriers and total spontaneous polarization as
a function of the polar distortion mode are reported. For the croconic acid, the height of the energy barrier is about 0.3 eV/unit
50
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
cell, similar to CBDC and PhMDA, consistent with the equivalent driving force for ferroelectricity. Indeed, as pointed out, the
OH · · · O bond lengths are almost identical for all compounds;
therefore, in materials in which the ferroelectric order is triggered
almost exclusively by the hydroxyl proton transfer, as CBDC, PhMDA and croconic acid, the energy differences between the polar
and centric phases are very similar. In the case of croconic acid,
as shown in Figure 13 and as we will demonstrate on the basis of
a symmetry-mode analysis, two inequivalent hydroxyl protons
together with a molecular buckling contribute to a total polarization of 24 µC/cm2 along the c-axis, the highest value found
so far within the class of organic ferroelectric compounds. For
2-FCHD, on the other hand, an active polar distortion lowers the
total energy to about 1 eV/unit cell, a value three times larger
than those for other materials. As shown in the bottom of Figure
13 by curved arrows, for 2-FCHD, two additive polar contributions must be taken into account together with proton transfer:
the tilting of carbon-fluorine bonds, and the non-planar deformation of the six-membered carbon rings for steric interactions,
which are energetically more favorable with respect to the molecular instabilities for CBDC, PhMDA and croconic acid, raising in
this way the energy barrier. Crystal symmetries impose the polar
axis to be in the ac-plane, with a magnitude of about 9.3 µC/cm2 .
4.3
SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
In order to shed light on ferroelectricity of these compounds,
we consider the relaxed structures of the high- and low-symmetry
phases, and analyze the displacive-type transition between the
two phases in terms of symmetry modes, using the Amplimodes
software package. In this way we can determine the global structural distortion that relates the two phases, enumerate the symmetry modes compatible with the symmetry breaking, and decompose the total distortion into amplitudes of these orthonormal symmetry modes.
Let’s start with CBDC and PhMDA. In Figure 14 we show their
centric structures, with the characteristic atomic displacements
51
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
Figure 14: Displacement patterns (arrows) connecting centric to polar
structures for atoms belonging to specified Wyckoff positions
(top to bottom) for CBDC (left) and PhMDA (right).
52
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
of the different polar distortion modes shown by coloured arrows. As the polar modes act separately on different Wyckoff
positions (WPs) of the high symmetry structure, it is meaningful to consider the action of polar distortion on atoms belonging
to different WPs separately. We denote these as A(WP), which
are reported from top to bottom in Figure 14. For CBDC, A(4e)
and A(4b) describe intra- and inter-molecular proton transfer distortions, and A(8f) a molecular buckling. For each of them we
calculated the polarization by displacing only the atoms belonging to a given WP orbit and keeping the rest of the ions in
their centrosymmetric positions, obtaining P4b = (6.6, 0, −5.5),
P4e = (0.5, 0, −1.6) and P8f = (5.5, 0, 0.4) µC/cm2 . Their sum
is (12.6, 0, −6.7) µC/cm2 , which is almost equal to the total polarization Ptot = (12.7, 0, −6.6) µC/cm2 calculated from the total
distortion. The linear addition of partial polarizations, shown
also in the left panel of Figure 15, is compatible with a displacivetype ferroelectricity.
Figure 15: Vectorial sum of partial modes polarizations for CBDC (left)
and PhMDA (right). Black arrows refer to partial polarizations labeled with their WP orbits, red and blue arrows refer
to the partial polarizations sums and total polarizations respectively.
In order to gain insights into ferroelectricity of CBDC, we note
how the mode decomposition shows that molecular buckling
gives almost as large a contribution as the inter-molecular proton
53
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
transfer in determining the total polarization. This effect can be
related to a double(π)-bond switching of carboxylic C = O ⇐⇒
C − O bonds correlated with the intermolecular and intramolecular hydrogen distortion. This is shown in Figure 16 where we
report a zoom of the A(8f) mode. In the upper part, we report
Figure 16: Switching between double and single bonds in the A(8f)
mode. Top part: centric phase; bottom part: switching between +P and −P. The chemist convention for the orientation of the dipole moment is used here, e.g. arrow starts at
δ+ and ends at δ−.
the centric structure, with hydrogens equidistant from nearest
carbons or oxygens, while in the lower part the cooperative hydrogen distortions leading to +P and −P state, which correlates,
in turn, with the switching of double and single carboxylic bonds
and with their contraction/elongation, in both +P and −P. This
is further confirmed by the following computational experiment:
(i) we first consider all the atoms at their centric positions (upper
part in Figure 16); (ii) we then move only the intramolecular hydrogen as, for instance, in +P state, keeping all the other atoms
fixed. The charge density difference between (i) and (ii) shows
54
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
an incipient pile up of out-of-plane charge between C2 = O2
and C1 = O3 , which corresponds to the initial formation of the
π (double) bonds. In Figure 17, the formation of such an outof-plane charge is in correspondence of C = O double bonds
(see again right-bottom panel of Figure 16). One expects that polar carboxylic groups rather than less polar C − C bonds might
be responsible for the large polarization in the buckling mode.
To confirm this, we have further decomposed the A(8f) mode
into contributions from the C − C and C = O bonds switching, and we found that P8f (C − C) = (0.85, 0, 0.43) µC/cm2 and
P8f (C = O) = (4.79, 0, −0.1) µC/cm2 . Again, their linearity is
fulfilled because P8f (C − C) + P8f (C = O) ≈ P8f (C − C + C = O)
= (5.5, 0, 0.4)µC/cm2 , and this clearly explains the origin of the
surprisingly large polarization of the buckling mode.
Figure 17: Difference between charge density isosurfaces of structures
reported in the upper and right-bottom parts of Figure 16. In
correspondence of C = O double bonds (the same of rightbottom panel of Figure 16), there is a formation of out-ofplane charge.
For PhMDA, we find through the Amplimodes software package three different partial modes, namely A(4a), A(4c) and A(8d),
whose contributions to the polarization are 5.8, 1.0 and 0.3 µC/cm2
along the z polar axis, respectively. The linearity holds also in this
case, but in PhMDA the intermolecular proton transfer does give
the dominant contribution, as shown in the right panel of Figure
15.
55
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
Figure 18: Displacement patterns (arrows) connecting centric to polar
structures for atoms belonging to specified Wyckoff positions
for the croconic acid.
A similar study can be carried out for croconic acid and 2FCHD. As for the former, in Figure 18 we report how the different
polar modes act on atoms belonging to different WP orbits. The
symmetry analysis reveals that the total distortion can be decomposed into four polar modes [A(4c), A(4b), A(4d), A(8e)], which
contribute to the total polarization along the z-axis by P4c = 10.0,
P4b = 8.8, P4d = 0.5 and P8e = 6.0 µC/cm2 , respectively. They
56
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
couple almost linearly to give a total polarization of 25.3µC/cm2
(the global distortion gives a total polarization of ∼ 24µC/cm2 ). It
is evident how the largest contributions come from inter- and intramolecular proton transfers; nevertheless, the polar mode A(8e)
contributes with a significant weight to the total polarization, although its polar distortion amplitude is about 5 times smaller
than that of proton transfer, as schematically shown in Figure 19
by arrows’ lengths. As in CBDC, the high value for the molec-
Figure 19: Polar distortion amplitudes for the croconic acid molecular
unit. Relative distortion differences are proportional to arrows’ magnitude differences.
ular buckling polarization has its origin in the double(π)-bond
switching of the polar carboxylic C − O bonds, suggesting that
this mechanism can be seen as a general property of the buckling
polar mode.
So far, we have analyzed three different types of polar modes
that independently contribute to the total polarization in hydrogen bonded organic ferroelectrics, namely the inter- intramolecular proton transfers and the molecular buckling. The 2-FCHD
compound allows to study a new mechanism that acts on materials with highly polar bonds, such as the carbon-fluorine C − F.
This mode, that in Figure 20 is labeled with A(4e), tilts the C − F
bond with respect to the centrosymmetric situation; the tilting
angle is about 6.9◦ . The other two polar modes, A(4a) and A(8f),
are the usual intermolecular proton transfer and the molecular
buckling. Their contributions to the total polarization are P4e =
(3.6, 0, 2.8), P4a = (−13.2, 0, −8.1) and P8f = (3.8, 0, 4.0) µC/cm2 ,
57
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
Figure 20: Displacement patterns (arrows) connecting centric to polar
structures for atoms belonging to specified Wyckoff positions
for the 2-FCHD. For the A(8f) mode, only distortions on oxygen atoms are shown.
58
4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY
respectively. It is possible to note that differently from CBDC,
PhMDA and croconic acid, for 2-FCHD the linearity of partial
mode doesn’t hold, because their sum gives a total polarization
of P4e+4a+8f = (−5.8, 0, −1.3) µC/cm2 , which is very different
from the polarization Ptot = (−8.3, 0, −4.2) µC/cm2 (for λ = 1)
calculated with all polar modes active. A possible reason for the
disagreement can be found in the fact that for 2-FCHD modes
A(4a) and A(8f) are very much correlated one to the other (they
are not independent), and cannot be analyzed separately. To justify this assumption, we carried out ab-initio calculations of partial polarization associated with the distortion A(4a+8f), and we
found that P4a+8f = (−12.1, 0, −5.9) µC/cm2 . In this way, linearity with mode A(4e) is almost recovered, obtaining P4e+4a+8f =
(−8.5, 0, −3.1) µC/cm2 , a value quite similar to the total polarization.
Finally, we calculate the dimensionless Born (or dynamical or
infrared) charge tensor (formula 2.37). In the extreme ionic limit,
Born charges coincide with static charges of the ions, giving nominal values. In a real material, Born charges account for dynamic
electronic polarization as well. In perovskite ABO3 oxides, the
ferroelectric tendency is well known to be connected with the
presence of anomalously large Born charges [52]. It should be
noted that in low symmetry cases, as in the present study, the
Born tensor is not symmetric in its Cartesian indices. Therefore,
we have split the tensor into symmetric Z∗S and antisymmetric
Z∗AS parts. In the following, we will focus on the former, and in
particular, on its three eigenvalues λ1 > λ2 > λ3 . Furthermore,
only the relevant active hydrogen Born tensors will be considered. We have also calculated the phonon frequencies at the Γ
point; the presence of an imaginary frequency usually implies a
structural instability, in this case of the paraelectric structure.
Let us first consider the CBDC. As expected, the significant
deviations of the dynamical tensor with respect to the nominal
charges involve the active H atoms. For the intermolecular hydrogen λi,Z∗S = (3.4, 0.4, 0.1), and for the intramolecular hydrogen
λi,Z∗S = (2.2, 0.3, 0.3). The large values of the Born charges for
hydrogens confirm their important contribution to the polarization. The other hydrogens have only negligible absolute eigenvalues ∼ 0.1. For the phonons, we found a large not-degenerate
imaginary Γ phonon frequency of about 106 cm−1 . According
59
4.4 CONCLUSIONS
to symmetry analysis [73], infrared irreducible representations
(namely Au and Bu ) exist for all three WP orbits. This is not
unexpected, as all WPs carry a contribution to the polarization.
In particular, the eigenvector of the imaginary frequency transforms under the symmetry operations of irreducible representation Bu , which is polar. After normalization to 1 Å, we use the
Amplimodes software to study the corresponding displacement
pattern. The largest absolute |u|, where u is the displacement of
the atom according to the phonon eigenvector, is 0.29 and 0.35 Å
for intra- and intermolecular hydrogens, respectively, Again, this
confirms the dominant role of the two types of hydrogen in the
ferroelectric properties.
For PhMDA, we found significant deviations with respect to
nominal values of the dynamical charge tensor for intermolecular hydrogen, whose eigenvalues are λi,Z∗S = (4.07, 0.40, 0.23).
The eigenvalues for other hydrogen atoms are smaller than the
nominal value 1. The imaginary phonon frequency for PhMDA
is equal to 112 cm−1 , again with polar symmetry Bu . Also in
this case, the polarization vector of the eigenmode has a large
displacement for the intermolecular hydrogen of ∼ 0.43 Å.
When performing similar calculations for the croconic acid and
2-FCHD, we noted that anomalous large Born effective charges
for hydrogen atoms active in the proton transfer mechanism occur also in these cases. We found, as larger Born effective charge
tensor eigenvalues λ1,Z∗S , 3.7 and 3.9 charge unit, respectively for
the croconic acid and 2-FCHD. Furthermore, in CBDC, PhMDA,
croconic acid and 2-FCHD very similar λ1,Z∗S values are consistent with the very similar OH · · · O bond lengths. This highlights
the fact that proton transfer gives a large contribution to the total
polarization as a common signature in all compounds.
4.4
CONCLUSIONS
In this chapter, we choose four prototypical organic molecular crystals, namely CBDC, PhMDA, the croconic acid and 2FCHD, which fulfill the necessary conditions required to host
ferroelectricity. We studied their ferroelectric properties through
DFT-based calculations, and found a spontaneous polarization as
60
4.4 CONCLUSIONS
large as 14.3, 7.0, 24 and 9.3 µC/cm2 , respectively. To shed more
light into their ferroelectricity, we analyzed the polar distortions
in terms of symmetry-adapted modes, and we classified the relevant polar mechanisms at play into four classes: intra- and intermolecular proton transfers, molecular buckling, and tiltings of
highly polar bonds. In all compounds, the proton transfer is
characterized by a very large value of the Born effective charge
associated to the active hydrogen atoms, and we can regard this
as a general property.
Furthermore, the increasing demand in organic electronics for
materials with advantages from the point of view of weight, flexibility, costs and environmentally-benign characteristics, pushes
materials science towards new design criteria and investigation
methodologies, in particular in the growing field of organic ferroelectrics. Our study might be regarded, in this sense, as one
of the first works in this direction, aiming at providing general
guidelines. Furthermore, we have demonstrated unambiguously
how the knowledge of symmetry properties of a given structure,
and the analysis of its polar modes through group theoretical
studies, are useful tools to interpret microscopically the polarization mechanisms.
By focusing on ferroelectric CBDC, PhMDA, croconic acid and
2-FCH, we have investigated four different polar mechanisms
that cooperatively contribute to the total dipole moment of these
molecular crystals, summarized in Table 2. A question spontaneously arises: is it possible to design a material where the desired polar modes are active? The answer is not so simple, because in general it is very difficult to couple different organic
functional groups showing desired polar activities into a unique
CBDC
PhMDA
croconic
2-FCHD
intra
√
-
inter
√
√
√
√
buckling
√
√
√
tilting
√
Table 2: Summary table of active polar modes for each organic ferroelectric studied.
61
4.4 CONCLUSIONS
compound. It is much more simple, at least from a theoretical
point of view, to control polarization by changing atomic species
inside the molecular units. We present here a simple example.
Let’s consider 2-FCHD; as we have seen, the tilting of the carbonfluorine C − F bonds strongly influences the total dipole moment.
It seems possible to change the contribution of this polar mode
by substituting the fluorine atom with chlorine, bromine or iodine atoms, for example. Bond lengths will be modified, and,
consequently, the dipole moment as well.
To conclude this chapter, it is important to highlight that, at
present, it is very difficult to completely characterize ferroelectricity in organic ferroelectric materials by means of first-principles
or group theoretical analysis only. This is due to the fact that it is
difficult to find general properties, such as energy barrier heights
or distortion amplitudes, that uniquely characterize a given polar mode. However, some common features can be highlighted.
In this work, for example, we found that a large Born effective
charge can be regarded as a proton transfer’s general property.
Furthers steps should therefore be carried out in the near future
in this fascinating field of organic ferroelectrics engineering, i.e.
one of the most promising area of modern research for device
applications.
62
5
P O S T- D F T S T U D Y O F C R O CONIC ACID PROPERTIES
The main purpose of this chapter is to present a comprehensive study of the croconic acid based on density functional theory using different levels of approximation for the exchange correlation functional, ranging from standard local and semilocaldensity functionals, such as Local-Density-Approximation (LDA)
and Generalized-Gradient Approximation (GGA) in the PerdewBurke-Ernzerhof (PBE) parametrization, to more advanced hybrid functionals such as Heyd-Scuseria-Ernzerhof (HSE) screened
hybrid functional and van der Waals type functionals, such as
Grimme’s functional (VdW-G). We will address the structural
and electronic properties of the croconic acid by testing the performances of different approximations with respect to available
experimental data. In particular, we will focus on the description
of the ferroelectric polarization provided by the approximations
used.
5.1
STRUCTURAL PROPERTIES
The use of different exchange-correlation functionals leads to
distinct ground Born-Oppenheimer surfaces, and therefore, to
different equilibrium configurations of the ions and, in turn, to
variations in the corresponding structural properties. In TABLE
3, we report equilibrium distances and angles for the used functionals, compared with available experimental data. The notation
is consistent with labels in the upper side of Figure 21. Larger
differences are involved in bonds between oxygen and hydrogen atoms, which give important contributions to the total polarization of the crystal. For this reason, the polarization is rather
strongly influenced by the approximation used.
63
1.300
1.293
1.230
1.241
1.249
1.419
1.472
1.499
1.487
1.453
1.048
1.052
127.35
122.03
1.283
1.273
1.225
1.240
1.247
1.423
1.462
1.482
1.472
1.451
1.089
1.099
125.98
122.30
PBE
126.91
121.64
1.297
1.287
1.216
1.218
1.229
1.397
1.461
1.498
1.489
1.443
1.012
1.012
PBE0
126.90
121.68
1.296
1.287
1.216
1.218
1.229
1.397
1.462
1.498
1.489
1.443
1.012
1.012
HSE
B3LYP
Å
1.303
1.295
1.218
1.218
1.232
1.397
1.462
1.502
1.494
1.445
1.013
1.009
deg
128.23
122.61
127.54
122.04
1.302
1.295
1.231
1.240
1.249
1.419
1.474
1.502
1.490
1.453
1.046
1.048
VdW-G
128.9(4)
122.1(4)
1.295(4)
1.306(4)
1.233(5)
1.213(4)
1.205(2)
1.382(5)
1.442(3)
1.508(6)
1.518(6)
1.466(6)
0.933
1.002
X-ray
128.7(3)
121.5(3)
1.285(9)
1.310(6)
1.201(6)
1.207(6)
1.257(8)
1.400(9)
1.489(7)
1.521(6)
1.481(8)
1.446(5)
1.022(8)
1.310(6)
neut.scat.
128.34(12)
122.32(11)
1.3020(4)
1.2964(6)
1.2141(3)
1.2160(6)
1.2292(4)
1.3980(4)
1.4691(5)
1.5105(5)
1.4974(4)
1.4449(6)
0.92(2)
0.849(12)
synchr.rad.
Table 3: Relevant structural parameters for the croconic acid molecule (bond lengths and angles) obtained from LDA,
GGA (PBE), PBE0, HSE, B3LYP and VdW-G calculations. Experimental values from X-ray [74], neutron scattering [75] and synchrotron radiation [75] experiments are also reported. Bonds’ notation in the first column is
consistent with that used in Fig. 21.
Bond
C1-O1
C2-O2
C3-O3
C4-O4
C5-O5
C1-C2
C2-C3
C3-C4
C4-C5
C5-C1
O1-H1
O2-H2
Angle
O1-C1-C2
O2-C2-C1
LDA
5.1 STRUCTURAL PROPERTIES
64
5.1 STRUCTURAL PROPERTIES
Figure 21: View of croconic acid’s unit cell along the a crystallographic
direction. In the molecule at the top-left of the unit cell, the
notation used in Table 3 is reported.
Variations in O-H bonds are about 10%, while in C-O and C-C
bonds they are as large as 2%. To compare the theoretical and experimental data, we calculated for each approximation the meanabsolute-deviation (MAD) and the root-mean-square (RMS) with
respect to each set of experimental results. Values are summarized in Table 4. MADs for bond lengths show that the gradient correction improves significantly the agreement. Each hybrid
functional gives almost similar MAD values, but smaller than
standard approximations. MADs calculated for VdW functional,
on the other hand, are very close to those for PBE. The same
behavior can be observed comparing the angles. The analysis
of RMS errors confirms the improvements carried on by hybrid
functionals. Thereby, we can argue that, despite their higher computational cost, hybrid functionals provide a better description of
the system’s structural properties.
65
5.2 ELECTRONIC PROPERTIES
MAD1
MAD2
MAD3
RMS1
RMS2
RMS3
LDA
0.044
0.042
0.059
0.056
0.066
0.083
PBE
0.031
0.042
0.044
0.038
0.075
0.062
PBE0
0.021
0.045
0.037
0.028
0.082
0.051
HSE
0.021
0.045
0.037
0.028
0.082
0.051
B3LYP
0.019
0.045
0.035
0.027
0.084
0.050
VdW-G
0.029
0.042
0.043
0.037
0.076
0.060
Table 4: Bond lengths mean absolute deviations (MADs) and root-meansquare (RMS) errors are calculated for each functional with respect to experimental data reported in Table 3. Labels 1, 2 and
3 refer to MAD and RMS calculated with respect to X-ray [74],
neutron scattering and synchrotron radiation data, respectively.
Among hybrid functionals, PBE0 and HSE give almost the
same results, as already known from literature [42]. With respect to local and semilocal approximations, these hybrid functionals generally tend to localize the electronic charge and result in stronger chemical bonds. Double bonds C1=C2, C3=O3,
C4=O4 and C5=O5 are shortened in hybrid calculations with respect to LDA, PBE and also VdW by about 0.02Å.
As for angles, all functionals are in good agreement with experimental data as for O2-C2-C1 angle, whereas somewhat larger
deviations are observed for the O1-C1-C2 angle (especially for
LDA). Finally, B3LYP hybrid functional better reproduces the experimental results.
5.2
ELECTRONIC PROPERTIES
In Figure 22 we report the croconic acid density of states (DOS)
calculated within LDA, PBE, van der Waals VdW-G and HSE approximations. We see that the introduction of density gradient
or van der Waals corrections doesn’t introduce significant modifications with respect to the standard local density approximation.
The energy gap between the top of the valence band and the bottom of the conduction band is about 1.5 eV for LDA, PBE and
VdW functionals. However DFT, both in local and semilocal ap-
66
5.2 ELECTRONIC PROPERTIES
proximations, underestimates the value of the energy band gap,
as well known. The introduction of a fixed amount of Fock exact
Figure 22: From top to bottom, the croconic acid density of states (DOS)
as calculated through LDA and PBE exchange-correlation
functionals, VdW and HSE hybrid functionals. In PBE and
HSE density of states, contributions from oxygens and carbons atoms are reported in blue and red dotted lines, respectively. For LDA and VdW functionals, we don’t expect significant differences. Origins of energy scales are set on Fermi
levels.
exchange by the HSE hybrid functional (bottom panel of Fig. 22),
gives rise to a downward shift of occupied states, because, for
localized oxygen and carbon orbitals, the exact exchange corrects
the self interaction. The energy gap opens up to about 2.7 eV
as indirect effect of the Coulomb potential screening (screening
acts in such a way to reduce the attractive Coulomb field). It’s
evident in Fig. 22 how the density spike in proximity of the
67
5.2 ELECTRONIC PROPERTIES
Fermi level in LDA, PBE and VdW approximations, (since it
presents a strong localized oxygen p-orbitals character) moves
towards deeper states in HSE. More precisely, we verified (Figures 23-24) that the PBE band decomposition of charge density,
and the projected density of states, show a strong pz character for
the O3-type oxygens in proximity of the Fermi level, and minor
contributions coming from hydrogen-bonded O4- and O5-type
oxygens; oxygen atoms covalently bonded to hydrogens almost
don’t contribute. A similar shift is then observed for the peak at
about −2.5 eV in the PBE DOS. Furthermore, in HSE DOS, states
are more localized, and in the energy range from −2.7 eV to the
Fermi level, three energy band gaps open with respect to the
one of LDA, PBE and VdW (Figure 22). In Figure 24, we also
Figure 23: Projected charge density in the range of energy [−0.5, 0] eV
immediately below the Fermi level. A local c-axis is set to
highlight the pz orbital of O3-type oxygens.
highlight a change in the spectral dispersion when going from
PBE (or LDA or VdW-G) to HSE approximations. Furthermore,
68
5.2 ELECTRONIC PROPERTIES
Figure 24: PBE and HSE projected density of states for oxygen atoms
in the energy range of 1 eV below the Fermi level. In the
density spike in proximity of Fermi energy, the separation of
contributions from different oxygen atoms is highlighted by
different colors.
the DOS character in the energy range [−0.5, 0] eV immediately
below the Fermi level is almost given by strongly localized pz
orbitals of O3-type oxygens in PBE, while it is characterized by
more delocalized and broad states in HSE.
As for the ferroelectric polarization, in the PBE approximation,
we find a net dipole moment as large as 24 µC/cm2 directed
along the c-axis, the larger value nowadays found within the
class of organic ferroelectric compounds, and in good agreement
with 26 µC/cm2 [71] calculated within the LDA approximation.
Furthermore, with the VdW correction, we find a total polarization of 25 µC/cm2 , very close to LDA and PBE calculations, and
69
5.2 ELECTRONIC PROPERTIES
to experimental results. With the use of HSE hybrid functional,
the polarization rises up to 30 µC/cm2 . We see that differences
between HSE and PBE polarizations come from a different distortion amplitude of the polar mode. For HSE, the total distortion
amplitude is ∼ 1.29Å, larger than that for PBE (∼ 0.73Å). In Figure
25, we show the difference vectorial field obtained as difference
between HSE and PBE relaxed positions. Oxygen and carbon
ions present displacements of about 0.12 and 0.07 Å respectively,
and HSE acts in such a way to contract covalent O-H and double C=O polar bonds. This strongly influences the ferroelectric
polarization.
Despite the fact that HSE better describes structural and electronic properties with respect to LDA, PBE and VdW-G, it gives
an overestimation of the ferroelectric polarization with respect to
the experimental value, as already noted for other simpler compounds [76]. The verdict on which functional to prefer - at least
in the study of croconic acid - is certainly still open.
Figure 25: Polar distortion introduced by the HSE hybrid functional on
the PBE relaxed structure. Arrows represent the difference
vectorial field obtained as difference between HSE and PBE
relaxed positions.
70
6
M U LT I F E R R O I C I T Y I N A M A N GANESE BASED MOF
Electricity and magnetism were unified into one common discipline in the 19th century by Maxwell equations. However, electric and magnetic ordering in solids are most often considered
separately: charges of electrons and ions are responsible for electric effects, whereas electron spins govern magnetic properties.
There are, however, cases in which these degrees of freedom are
strongly coupled. For example, the effects of spins on charge
transport properties of solids, giving rise to giant magnetoresistance phenomena - awarded by the Nobel Prize in Physics 2007
to A. Fert and P. Grünberg [77] - are at the basis of the important
field of spintronics, a new branch of electronics which aims to exploit the spin as well as the charge of electrons for new electronic
devices. From a historical point of view, the linear coupling between magnetism and electricity, which would cause, for example, a magnetization proportional to an electric field, was first
predicted and then experimentally observed by Dzyaloshinskii
[78] and Astrov [79] respectively. This coupling is now known as
linear magnetoelectric effect. A step further is that not only the
appearance of magnetization M in an electric field E or the inverse effect of electric polarization P generated by the application
of magnetic field H can exist in solids, but that there may exist
systems in which two types of ordering, e.g. (ferro)magnetism
(the spontaneous ordering of spin magnetic moments) and ferroelectricity (the spontaneous ordering of electric dipole moments),
can coexist in one material in the absence of external electric and
magnetic fields. After Schmid [80], these materials are called
multiferroics, and their great potential for practical applications
has lead to an extremely rapid development of the research in
this field. Applications include the ability to electrically control
magnetic memories, the creation of new devices based on 4-state
71
MULTIFERROICITY IN A MANGANESE BASED MOF
logic (e.g., with both up and down polarization and up and down
magnetization) and magnetoelectric sensors.
6.0.1
Different types of multiferroics
Most of the many new multiferroics that have been discovered in the last few years are transition metal oxides. The possible mechanisms that could lead to multiferroicity in these compounds can be divided into four classes [81] (see Figure 26): a)
lone-pair effects, as in BiFeO3 and BiMnO3 , where lone pairs
of electrons on the A cation in the perovskite structure distort
the geometry of the anion cage, resulting in ferroelectricity; b)
geometric frustation, as for YMnO3 , in which long-range dipoledipole interactions and distortion of oxygen octahedra generate
a stable ferroelectric state; c) charge ordering, i.e. where noncentrosymetric arrangements of mixed-valent ions induce ferroelectricity in magnetic materials, as in LuFe2 O4 ; d) magnetic ordering, as for T bMnO3 , DyMnO3 and T bMn2 O5 , in which ferroelectricity is induced by magnetic long-range order, i.e. the
arrangement of magnetic dipoles doesn’t show inversion symmetry.
6.0.2
Metal-organic frameworks
Very recently, the search for new multiferroics and ferroelectrics
has been extended to include organic compounds [71, 83]. Among
them, we highlight in particular materials known as metal-organic
framekorks (MOFs), crystalline compounds consisting of metal
ions coordinated by organic molecules [84]. MOFs have long
been studied since they provide an impressive number of applications (such as gas storage, catalysis and drug delivery [85, 86]),
but have not been considered for multiferroic purposes until recently [5]. Particularly interesting are MOFs with the perovskite
ABX3 architecture, some of which present a multiferroic behavior.
The exciting properties of this new class of materials for device
applications [87, 88] come from their hybrid nature, benefiting
from the characteristics of both the inorganic and organic building blocks. One of the main advantages of magnetic MOFs is the
possibility to control the nature of magnetic coupling by modify-
72
MULTIFERROICITY IN A MANGANESE BASED MOF
Figure 26: a) In materials like BiFeO3 the ordering of lone pairs (yellow lobes) of Bi3+ ions, contributes to the polarization (green
arrow). b) The tilting of a rigid MnO5 block represents the
geometric frustation mechanism that generates ferroelectricity
in YMnO3 . c) In charge ordered systems like LuFe2 O4 , the
coexistence of inequivalent sites with different charges, and
inequivalent (long and short) bonds, leads to ferroelectricity.
d) In materials like T bMnO3 , Mn spins order so that the tip
of the spins sweep out a cycloid generating a polarization.
The figure is modified from [82].
ing the starting building blocks and by searching among the variety of possible metal ions, short-ligands, co-ligands, templates etc.
This flexibility, fundamental for materials science engineering, is
not so wide for inorganic compounds. Furthermore, in multifer-
73
6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF
roic MOFs there is also hope to control ferroelectric properties,
opening the way to new materials exhibiting high spontaneous
polarization.
Very recently, a series of novel perovskite-like metal-organic
frameworks have been successfully synthesized by Hu et al. [84].
Among them, the copper-based compound [C(NH2 )3 ]Cu(HCOO)3
is particularly interesting, and it was also the first multiferroic
MOF to be studied from a theoretical and computational point
of view with first-principles DFT-based calculations by Stroppa
et al. [89]. It is evident how, compared to inorganic compounds,
a great variety of A and X3 functional organic groups, in addition to B metal ions, can be taken into account for MOFs. In the
present study, we will carry out ab-initio simulations on the manganese based metal organic framework [CH3 CH2 NH3 ]Mn[(HC
OO)3 ] with the perovskite like structure ABX3 , where the organic groups A and X are the ethylammonium [CH3 CH2 NH3 ]+
and the carboxylate [HCOO]− respectively, and B is the divalent
Mn2+ ion. Of this manganese metal-organic framework (firstly
synthesized by Wang et al. [90] and to which we refer hereafter
as Mn-MOF), we will first investigate the structural properties
and the spin ordering; then we will analyze the ferroelectric polarization by using a distortion-mode analysis sheding light into
different mechanisms at play. In the general framework of hydrogen bonded compounds, we will investigate the role of weak
interactions in this ferroelectric Mn-based MOF.
6.1
CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF
The key feature of the structure of this Mn-MOF is the anionic
NaCl-framework of [Mn2+ (HCOO)−
3 ] where the nearly cubic
cavities in the perovskite are occupied and the charge counterbalanced by the ethylammonium [CH3 CH2 NH3 ]+ cations, as in
Figure 27. The compound belongs to the polar space group
Pna21 , so a spontaneous polarization is allowed by symmetry.
Each manganese ion is connected to its six nearest neighbors by
six bridging ligands, with octahedral coordination geometry and
Mn − O distances in the range 2.17 − 2.23 Å. We started our cal-
74
6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF
Figure 27: Ball-and-stick model of the Mn-MOF. Manganese oxygen carbon nitrogen and hydrogen ions are represented with yellow
red green blue and black balls respectively. The closed dotted
line highlights one of the cavities in the perovskite structure
where the A group is accommodated. See also Figure 30.
culations from experimental crystallographic data [91], optimizing the structure until Hellmann-Feynman forces were not larger
than 0.02 eV/Å. Kohn-Sham equations were solved using the projector augmented-wave (PAW) method with the PBE functional,
and the energy cutoff for the plane wave expansion was set to
400 eV; furthermore, a 3 × 4 × 2 Monkhorst-Pack grid of k-points
was used. In the lattice, the Mn − O − CH − O − Mn linkages
are along the c-axis and along the two diagonal directions of the
ab-plane. The Mn · · · Mn distances via bridging HCOO− are in
the range 6.03 − 6.16 Å. In the ethylammonium cation, the NH3
and CH3 groups form both N − H · · · O and C − H · · · O weak
hydrogen bonds to the oxygens atoms of the framework. The
octahedrally coordinated Mn2+ ions (d5 with t32 e2 electronic
configuration) lies approximately at the center of a slightly distorted octahedron with two short (s) and two long (l) equatorial Mn − Oeq bond lengths (2.17, 2.18 Å and 2.21, 2.23 Å respec-
75
6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF
tively) and one short and long apical Mn − Oap (2.17 and 2.22 Å).
The octahedra are tilted about the c-axis by ∼ 30◦ . The Mn-MOF
can then be viewed as composed of chains running along the
c-axis; within each chain, the Mn units are connected by apical
HCOap Oap groups, whereas parallel chains are linked by equatorial HCOeq Oeq groups. Within the ab-plane, the Mn − Oeq
units display an alternate pattern of elongated axes, defined by
the long Mn − Oeq bonds, and shortened axes, defined by the
short Mn − Oeq bonds (see left panel of Figure 28). This situation is similar to the antiferro-orbital ordering [92] on Cu sites in
the well-studied KCuF3 perovskite [93] and in the above cited
copper based metal-organic framework [89]. In these two lat-
Figure 28: ab-plane (left) and along c-axis (right) views, and corresponding long (l) and short (s) bond lengths. The organic
A groups are not displayed inside the perovskite cavities.
ter cases, however, the in-plane antiferro-distortive pattern is due
to the strongly Jahn Teller active transition metal Cu2+ . In the
Mn-MOF, on the other hand, the divalent Mn doesn’t show any
Jahn Teller effect, and the distortion rather seems to be a consequence of the strong√orthorhombic unit cell, being a/b = 1.11 6= 1
and c/a = 1.32 < 2. In order to demonstrate this, we carried
out the following computational experiment: we forced the compound into √
a standard cubic perovskite structure, i.e. a/b = 1
and c/a = 2, keeping the volume constant; after ionic relax-
76
6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF
ations, long and short Mn − Oeq bonds differ only by 0.02 Å,
a much smaller difference than that for the orthorhombic perovskite structure. In this way we demonstrate that, at the origin
of octahedra distortions in Mn-MOF, there is not an electronic effect (as the Jahn Teller), but instead a lattice-induced or steric one.
Along the chain, the crystal shows a consecutive pattern of short
and long Mn − Oap bonds (right panel of Figure 28). These different lengths are mainly due to hydrogen-bonds that H atoms of
the A groups form with the equatorial oxygens of the framework.
The importance of hydrogen-bonds will be investigated in light
of the symmetry-mode analysis for the microscopic mechanisms
at play in the polar distortion.
Furthermore, there is a direct coupling between the electronic
orbital filling and the magnetic structure. The Goodenough Kanamori Anderson (GKA) rules [94, 95] suggest that there is a strong
antiferromagnetic coupling if on corresponding sites the half-occupied orbitals are directed towards each other. In our case, divalent Mn sites have all d orbitals singly-occupied which direct
towards each other and interact via a superexchange mechanism
mediated by the molecular orbitals of the HCOO ligand groups.
Thus, the Mn sites are coupled antiferromagnetically (AFM) both
in the ab-plane and along the c-axis, so that the ground state
displays a G-type AFM spin configuration, as shown in Figure
29. Furthermore, we found that the magnetic moment associated
with each Mn ion is about 4.5µB , consistent with the d5 electronic
configuration. We have also performed calculations for C- and
A-type spin configurations, finding an increase in energy with
respect the G-type ground state of 0.013 and 0.025 eV/unit cell,
respectively.
Mapping the system onto a Heisenberg model H =
P
− ij Jij Si Sj , we find for the intra- and inter-chain coupling constants (see Figure 29) Jc ≈ −4 and Ja ≈ −8 cm−1 respectively,
with Si Sj = ±25/4 (+ or − depending on the relative orientations of spins in the i and j sites). Since negative values for
the exchange constants refer to antiferromagnetic couplings, our
estimated exchange coupling constants are in agreement with
the GKA rules. Moreover, for the Mn-MOF it is energetically
more convenient to reverse the reciprocal direction of two nearest
neighbors spins along the c-axis than two spins in the ab-plane.
77
6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION
Figure 29: G-type AMF spin ordering for the Mn-MOF. Jc and Ja refer
to the exchange coupling constants of the Heisenberg model.
6.2
MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION
As shown in chapter 4, a pseudosymmetry analysis is a useful
tool for analyzing the microscopic origin of ferroelectric polarization. So, for the Mn-MOF, we can compare a virtual paraelectric
parent structure with the ferroelectric phase. We can see in Figure 30 that the mechanisms at play are essentially the distortion
of the organic A groups and the very small deformation of the
octahedra, as arrows and guidelines highlight. The main contribution to the total polarization comes from the distortion of
the A groups in the perovskite cavities. In the centric structure
(λ = 0), this group shows a planar configuration of the C − C − N
bonds with six hydrogen atoms symmetrically arranged outside
and two inside this plane. In the polar phase (λ = 1), the Ethylammonium molecule tilts; C − C − N bonds tilt and a net nonzero dipole moment is induced. Furthermore, distances between
the hydrogen atoms of the A group and the oxygens of the frame-
78
6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION
Figure 30: Ball-and-stick model of Mn-MOF in the centric (λ = 0) (left)
and polar (λ = 1) (right) phases. Most important mechanisms contributing to the total polarization are highlighted
by closed lines and arrows.
work range from 1.75 to 2.70 Å, with the formation of weak hydrogen bonds. The presence of organic groups, together with
the small tilting of the octahedra, reduces the space group from
the centrosymmetric Pnma to the polar Pna21 , with a maximum
atomic displacement between λ = 0 and |λ| = 1 of about 1.51 Å.
As expected, the polar phase is calculated to be more stable than
the centric one by about 0.38 eV/formula unit, a result five times
larger than that for the copper based MOF studied in [89]. This
is because in Cu-MOF, the mechanism which breaks the inversion symmetry, i.e. the correlation between the antiferro-orbital
distortive ordering and the hydrogen bonding between COOH
and A groups, introduces in the structure a maximum atomic
displacement of about 0.24 Å, much smaller than that for the MnMOF. The small displacement is also consistent with the weakness of the hydrogen bonding which ultimately induces ferroelectricity in Cu-MOF. In Mn-MOF, the total energy as a function
79
6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION
of the amplitude of the polar distortion λ, produces the expected
double-well profile characteristic of a switchable ferroelectric system (top panel of Figure 31), with a total spontaneous polarization of magnitude 1.64 µC/cm2 along the c-axis for λ = ±1 (bottom panel of Figure 31). The paraelectric phase (λ = 0) does
Figure 31: Variation of total energy (top) and of polarization (bottom) as
a function of the amplitude of the polar distortion between
centric (λ = 0) and polar (λ = ±1) configurations.
not represent any physical state for the structure, but it is a reference state with a zero dipole moment through which we build
the path to calculate the polarization of the ferroelectric state (we
recall that only polarization differences have a well-defined physical meaning). The paraelectric state is so high in energy with
respect to the polar state (about 1.5 eV/unit cell) because, to restore the inversion symmetry in the structure, we forced the A
group to be in a planar configuration, with an estimated energy
rising of ∼ 0.6 eV/isolated A group. We expect that the physical
80
6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION
mechanism through which Mn-MOF reverses the polarization is
an almost rigid rotation of the A group molecule as a whole when
an electric field is applied.
The global polar distortion can be decomposed following a
symmetry-mode analysis. For this purpose, we consider the
relaxed structures of the high- and low-symmetry phases, analyzing the displacive-type transition between the two phases in
terms of symmetry modes using the Amplimodes software package. In Figure 32 we show the centric structure for Mn-MOF
alone, with the atomic displacement field for different polar distortion modes highlighted by colored arrows. Since these polar
modes act separately on different Wyckoff positions of the highsymmetry structure, we can decompose the global distortion into
contributions coming from distortions of atoms belonging to different WPs orbits, as we did in the study of hydrogen bonded
organic ferroelectric. We find three different polar contributions,
labeled as A(4a), A(4c) and A(8d), which act on the manganese
ions, on the carbon, nitrogen and in-plane hydrogen atoms of the
A groups, and on the octahedral frameworks and out-of-plane
hydrogen atoms of A groups, respectively. For each of them we
have calculated the polarization by displacing only the atoms belonging to a given WP orbit, and keeping the rest of the structure
in its centrosymmetric positions, obtaining P4a = (0, 0, −0.37),
P4c = (0, 0, 2.08) and P8d = (0, 0, −3.61) µC/cm2 . Notably, their
sum, P4a+4c+8d = (0, 0, −1.90) µC/cm2 , is almost equal to the total polarization for λ = 1, being this linearity of partial polarizations compatible with the displacive-type ferroelectricity. However, while in compounds such as BiFeO3 the displacive-type ferroelectricity arises from the hybridization of empty orbitals of A
site atoms with the oxygen p-orbitals, and from the corresponding formation of a covalent bond, in Mn-MOF the ferroelectricity
has a different origin. It is due to the presence of polar A-groups
which are then coupled to oxygen atoms of B-groups via hydrogen bondings.
We also calculated the contributions to the total polarization
coming from different functional groups of the perovskite structure, distorting only atoms belonging to the functional group
under investigation and keeping the rest of the structure in its
centrosymmetric position. As mentioned above, the major contribution comes from the tilting of the A groups inside the cavities
81
6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION
Figure 32: Displacement patterns (arrows) connecting centric to polar
structures for atoms belonging to specified Wyckoff positions
(top to bottom) for Mn-MOF.
82
6.3 ELECTRONIC PROPERTIES
of the perovskite, with a net dipole moment of PA = (0, 0, −2.66)
µC/cm2 in the λ = 1 phase. However, the distortion associated
with the ligand groups X3 cannot be neglected (it contributes
with an opposite moment PX3 = (0, 0, 1.48) µC/cm2 ), as well as
the manganese ions that bring in a contribution PB = (0, 0, −0.37)
µC/cm2 . If we take their sum, we find a polarization PA+B+X3
= (0, 0, −1.55) µC/cm2 , in good agreement with that calculated
for the global polar distortion PABX3 = (0, 0, −1.64) µC/cm2 (bottom panel of Figure 31).
6.3
ELECTRONIC PROPERTIES
In the top panel of Figure 33, we report the projected density
of states (PDOS) for a Mn ion with a localized spin up moment.
We find an energy gap between valence and conduction bands
of ∼ 2 eV, and we highlight that all Mn d-orbitals are below the
Fermi level; this is consistent with the d5 electronic configuration
of divalent Mn2+ ions. The two spikes just below the Fermi energy can be identified with eg and t2g orbitals, even though the
lack of a perfect cubic symmetry and the octahedra tilting with respect to the z-axis produce the mixing of dxy , dxz and dyz bands
with d3z2 −r2 and dx2 −y2 ones. Nevertheless, the spikes’ separation ∆CF gives an approximated value for the crystal field, which
we find to be ∼ 1.2 eV. Furthermore, we note that Mn d-orbitals
are very narrowed and localized in energy, apart from a small
hybridization with oxygen orbitals around −4 eV. Simulations
were performed under the PBE approximation for the exchangecorrelation functional. A further approximation with respect to
PBE is the introduction of the U on-site Coulomb energy. The
LSDA+U [96–98] scheme can overcome some of the deficiencies
of LSDA. However, LSDA+U suffers ambiguities in the choice of
the U parameter, and needs a choice regarding which orbitals
to treat within a Hubbard-like approach. For simple materials,
a self-consistent evaluation of the U parameter can be obtained,
although this method is not widely used [99]. We used the Dudarev approach [100], where the parameters U and J (the on-site
exchange interaction parameter) do not enter separately, but only
their difference is meaningful. To the best of our knowledge, this
83
6.3 ELECTRONIC PROPERTIES
Figure 33: Density of states for a Mn ion with a localized spin up moment in PBE approximation (top panel) and with a Hubbardlike correction in the Dudarev approach Ueff =U-J=4 eV (bottom panel) . The black line refers to the total DOS (minority
spin density is shown as negative), while in blue and red
we report the majority and minority projected densities onto
spin-up Mn ion. The energy scale is with respect to the Fermi
level. Mn d-orbitals are fully occupied, consistently with the
d5 electronic configuration.
is the first attempt to describe MOF compounds in the framework of LSDA+U, so no comparison values for the U parameter
are available. We performed calculations varying Ueff =U-J in the
range from 1 to 4 eV. The introduction of the on-site Coulomb
energy rises the band gap up to ∼ 3.5 eV for Ueff =4 eV, and increase the hybridization of t2g orbitals with oxygen ones, leaving
eg bands almost invariant (see bottom panel of Figure 33). For
84
6.3 ELECTRONIC PROPERTIES
Ueff ∼ 2 eV, the t2g spike disappears in favor of more delocalized
states.
Figure 34: From top to bottom, the PBE total density of states and the
PBE density of states for each group of the ABX3 perovskite
structure are shown. The energy scale is with respect to the
Fermi level.
In Figure 34 we show the density of states for each group of the
ABX3 perovskite structure calculated in the PBE approximation.
We highlight that only negligible magnetic moments are induced
by Mn ions on A and X groups atoms, so in Figure 34 we show
only one spin channel. As for A groups states, we note that they
are more localized in energy because, apart for weak hydrogen
bonds with the oxygen atoms of the framework, in first approximation, they can be considered isolated inside the perovskite
cavities. Therefore, hydrogen bonds produce a small broadening
only at ∼ −4 eV (with respect to the Fermi level) with atoms of
the ligand X groups.
85
7
CONCLUSIONS
In this work we have presented a comprehensive study of four
hydrogen-bonded molecular crystals ferroelectric properties. Using first-principles DFT-based calculations and symmetry analysis methods, we have studied the origin of ferroelectricity in
CBDC, PhMDA, croconic acid and 2-FCHD. DFT simulations
were performed within the generalized gradient approximation
(GGA) for the exchange-correlation functional as proposed by
Perdew, Burke and Ernzerhof (PBE). Hybrid functionals such as
HSE and Grimme’s correction to the standard PBE (vdW-G) - to
take into account weak van der Waals interactions - were used to
go beyond the local and semilocal approximations.
For all compounds, our study started from a pseudosymmetry analysis, in which a given low-symmetry (ferroelectric) structure is represented in terms of a symmetry-lowering Landau-type
structural phase transition from a high-symmetry (paraelectric)
parent structure, i.e. a given ferroelectric structure is represented
in terms of a ferroelectric distortion from a parent paraelectric
phase. CBDC, PhMDA, the croconic acid and 2-FCHD crystallize
respectively in the polar Cc, Pna21 , Pca21 and Cc space groups,
and we found that their pseudosymmetric centric structures have
space group symmetry C2/c, Pbcn, Pbcm and C2/c. The estimated spontaneous polarizations are as large as 14.3, 7.0, 24 and
9.3 µC/cm2 , respectively. Furthermore, we have shown that a
partial symmetry-mode analysis is a useful tool to explore the
polar mechanisms at play. In all molecular crystals, the proton
transfer between molecular units (sometimes called intermolecular proton transfer) appears to give the main contribution, as confirmed by the large dynamical charges (or dimensionless Born
effective charges) and by the analysis of the eigenmode displacement patterns. We found that the intermolecular proton transfer
carries a contribution of about 8.6, 5.8 and 10 µC/cm2 for CBDC,
PhMDA and croconic acid (for 2-FCHD intermolecular proton
transfer is strongly correlated to a buckling polar mode), with a
86
CONCLUSIONS
Born effective charge associated to relative active hydrogen atoms
of 3.4e, 2.2e and 3.7e, respectively. Larger values of the Born effective charge with respect to the nominal ones are the fingerprints
of an active role of related atoms in the ferroelectric transition.
We have also calculated the phonon frequencies at the Γ point for
each paraelectric phase, and found imaginary frequencies. The
presence of an imaginary frequency usually implies a structural
instability, in these cases of the paraelectric structures towards
the ferroelectric phase. Despite the fact that proton transfer appears to give the main contribution, other polar modes such as
the π-bond switching of carboxylic groups in CBDC and croconic
acid associated to buckling distortions, or the tilting of polar C-F
bonds in 2-FCHD, may also have significant weights in the final
polarization.
As for the croconic acid, we also carried out a deeper study
of structural and electronic properties using different levels of
approximations for the exchange-correlation functional, ranging
from LDA and PBE density functionals, to HSE, PBE0, B3LYP
and vdW-G hybrid functionals. Structural properties were compared with available experimental data. We found that hybrid
functionals HSE, PBE0 and B3LYP, despite a larger computational
cost, provide a better description, both for bond lengths and for
angles. Furthermore, the Grimme’s corrected functional vdW-G
does not introduce larger differences with respect to the standard
PBE exchange-correlation functional. We also compared the density of states (DOS) as calculated through LDA, PBE, vdW-G and
HSE. As expected, LDA, PBE and vdW-G functionals give almost
the same results, with an energy band gap between valence and
conduction bands of ∼ 1.5 eV, and a strongly localized spike at
the Fermi level with oxygen pz -orbital character. The introduction of a fixed amount of exact exchange in the HSE hybrid functional, as a by-product of the self-interaction correction (SIC) and
the screening of long-range Coulomb interaction, opens the band
gap up to ∼ 2.7 eV, and produces a downward shift of localized
states. As for the ferroelectric polarization, PBE and vdW-G functionals give a total dipole moment of 24 and 25 µC/cm2 respectively, in agreement with the experimental result of 21 µC/cm2
[71]. With HSE, the polarization rises up to 30 µC/cm2 , resulting
from a larger total distortion amplitude of the polar mode with
87
CONCLUSIONS
respect to LDA, PBE and vdW-G (∼ 1.29Å for HSE and ∼ 0.73Å
for PBE).
A special class of new materials is that of multiferroics. Such
materials, which combine at the same time both a magnetic and
a ferroelectric order, are potentially very useful in the field of
spintronics. Most of the new multiferroics that have been discovered in the last few years are transition metal oxides. Very
recently, however, the search for new multiferroics and ferroelectrics has been extended to include organic compounds, such
as metal-organic frameworks (MOFs). Particularly interesting are
MOFs with the perovskite ABX3 architecture, some of which
present a multiferroic behavior. In this work we also carried
out a DFT study of a manganese based MOF. The key feature
of the structure of this Mn-MOF is the anionic NaCl-framework
of [Mn2+ (HCOO)−
3 ] where the nearly cubic cavities in the perovskite are occupied and the charge counter-balanced by the ethylammonium [CH3 CH2 NH3 ]+ cations. We found that in the ferroelectric state, the structure presents a total dipole moment of
1.6 µC/cm2 , whose origin is mainly due to the presence of the
polar organic molecule in the perovskite A site. As an induced effect, hydrogen bonds that A groups form with the oxygens atoms
of the framework produce a distortion of the octahedra which
gives rise to a finite polar contribution. As for the magnetic
order, the ground state is a AFM-G type spin configuration, in
agreement with Goodenough Kanamori Anderson (GKA) rules
2+ ion. Mapping the system onto a Heisenberg
for a divalent Mn
P
model H = − ij Jij Si Sj , we found Jc ≈ −4 and Ja ≈ −8 cm−1 ,
with Si Sj = ±25/4 (+ or − depending on the relative orientations
of spins in the i and j sites) and Jc and Ja representing the outof- and in-plane coupling constants between Mn spins. We also
investigated the electronic structure of Mn-MOF, finding a band
gap of ∼ 2 eV and a crystal field splitting ∆CF of ∼ 1.2 eV. The introduction of a Hubbard-like correction in the Dudarev approach
with Ueff =U-J=4 eV produces in this case a larger hybridization
of t2g states with ligands oxygens p-orbitals, and opens the band
gap up to 3.5 eV.
In conclusion, we hope that our study will stimulate further
attempts to search for new organic ferroelectrics and MOFs with
potentially large spontaneous polarizations and will give useful
guidelines for further works in materials design. We also demon-
88
CONCLUSIONS
strated that DFT-based calculations and group theoretical analysis are very useful investigation methodologies in this fascinating
field of modern materials engineering.
89
Part III
Appendix
90
A
A B O U T C O R R E L AT I O N S
A simple way to understand the origin of correlation energy is
to interpret it as the difference between the full ground-state energy obtained through the correct many-body wavefunction and
the one obtained within the Hartree-Fock approximation. Furthermore, since the correlation term arises from a more general
trial wavefunctions than a single Slater determinant (see for example A.2), it cannot raise the total energy, being Ec [n] 6 0.
Since a single Slater determinant is itself more general than a
simple product of indipendent one-electron orbitals, Ex [n] 6 0
must hold. For these reasons, the upper bond Exc [n] 6 0 can
always be verified.
Recalling the wavefunction’s quantum mechanical interpretation as probability amplitude, the many-body wavefunction written simply as the product of one-electron orbitals leads to consider the probability amplitude of the many-body interacting system as the product of the probability amplitudes of individual
non interacting one-electron systems. Mathematically, the probability of a composed event is equal to the product of individual
events’ probability only if they are independent, i.e. they are uncorrelated. Physically, electrons described by means of a product
wavefunction don’t interact, neglecting the fact that, as a consequence of Coulomb interaction, they try to avoid each other. For
this reason, the correlation energy is simply due to mutual repulsions of the interacting electrons.
A rather different, but equivalent, way to understand correlation is to consider the following alternative form for the Coulomb
interaction operator [101]:
V̂ee
e2
=
2
ZZ
drdr 0
n̂(r)n̂(r 0 ) − n̂(r)δ(r − r 0 )
|r − r 0 |
,
(A.1)
in which the operator character is carried by the density operator n̂(r), and the term with the delta-function subtracts out the
91
ABOUT CORRELATIONS
self-interaction term. In this formalism, the electron density n(r)
is just the expectation value of the density operator over the
distribution probability given by the many-body wavefunction.
Writing explicitly quantum density fluctuations by the ansatz
n̂(r) = n(r) + δn̂fluc , one can see that the expectation value of the
operator A.1 differs from the classical Coulomb interaction energy EHartree [n] just for the contributions coming from δn̂fluc
and the self-interaction term. Quantum fluctuations are thus at
the origin of quantum correlations between interacting electrons.
This can be emphasized looking for an explicit trial manybody wavefunction that contains correlations between electrons.
By definition, the Hartree-Fock solution ΦHF (r1 , · · · , rN ) for the
ground-state of a homogeneous electron gas doesn’t contain correlations, however, a simple way to incorporate them is to make
the following Jastrow ansatz [102]:


X
Ψ(r1 , · · · , rN ) = exp 
f(ri − rj ) ΦHF ,
(A.2)
i,j
where f(ri − rj ) is determined by energy minimization. By means
P
of the density fluctuation nq = i e−iqri , we can write the wavefunction A.2 in the form


X
Ψ(r1 , · · · , rN ) = exp 
τ(q)n†q nq  ΦHF .
(A.3)
q
The function τ(q) is the Fourier transform of f(r) and can be considered a variational function. The wavefunction A.3 consists of
an independent one-electron part ΦHF that takes into account the
exchange energy, and of an exponential prefactor which has the
2
form of independent harmonic oscillators’ ground-state (e−x ),
being oscillator variables proportional to nq . These density fluctuations can be regarded as variables associated to collective degrees of freedom of the electronic system, e.g. zero-point fluctuations of plasmons, describing mutual screening and long-range
correlations of electrons. By means of a trial wavefunction as
in A.2, D. Ceperley and B. Alder in 1980 studied the phase diagram of a homogeneous electron gas by Monte Carlo methods
[10]; from their studies, expressions like 1.15 have been derived
92
ABOUT CORRELATIONS
Figure 35: Diagrams represented the expansion of the self-energy
Σ̂(1, 2). The one-particle Green’s function G(1, 2) is represented by an arrow from 2 to 1, and the screened potential
W(1, 2) by a wiggly line between 1 and 2.
to parameterize the exchange-correlation functional in the local
density approximation.
Screening is the simplest and probably the most important effect due to correlations. This can be seen introducing the conceptual tool of the one-particle Green’s function [103]
i
G(1, 2) = − hT̂ Ψ(1)Ψ† (2)i
h
,
(A.4)
where 1 stands for the five coordinates of a particle: space, spin
and time, e.g. (1) = (r1 , σ1 , t1 ) = (x1 , t1 ); obviously the same
for 2. T̂ is the Dyson time-ordering operator and Ψ is the oneelectron field in the Heisenberg representation. Brackets refer
to the average operation with respect to the exact many-body
ground state. The Green’s function obeys the equation
[ε − ĥ(x) − V̂(x)]G(x, x 0 ; ε)
Z
− dx 00 Σ̂(x, x 00 ; ε)G(x 00 , x 0 ; ε) = δ(x − x 0 )
,
(A.5)
93
ABOUT CORRELATIONS
where
ĥ(x) = −
h2 2
∇ −
2me
all nuclei
X
Zn v̂(x, Rn )
n
Z
V̂(x) = dx 0 v̂(x, x 0 )ρ(x 0 )
v̂(x, x 0 ) =
e2
|x − x 0 |
ρ(x) = hΨ† (x)Ψ(x)i = −ihG(x, t; x, t + ∆) (∆ → 0+ )
Z
iε
G(x 00 , x 0 ; ε) = d(t − t 0 ) G(x, t; x 0 , t 0 ) exp
(t − t 0 ) .
h
Σ̂ is the self-energy operator and takes into account the complicated correlation effects of the interacting many-particle system.
A series expansion of Σ̂ in terms of the Coulomb interaction v̂
gives, as first term, the Hartree-Fock exchange potential
Σ̂HF (x, x 0 ; ε)
=
−v̂(x, x 0 )hΨ† (x 0 )Ψ(x)i
= ihv̂(x, x 0 )G(x, t; x 0 , t + ∆)
,
(A.6)
which is independent of ε. However, the expansion Σ̂ in a power
series in terms of v̂ is not a good solution, since such expansion
usually diverges for metals, and when it is convergent, its convergence rate rapidly drops as the polarizability of the system increases. These problems can be overcome through an expansion
in terms of a screened potential Ŵ rather than the bare Coulomb
potential v̂ [104]. The potential Ŵ was first introduced by Hubbard [105] in the following way:
Z
i
d(3)d(4) v̂(1, 3)hT̂ ρ 0 (3)ρ 0 (4)iv̂(3, 4)
Ŵ(1, 2) = v̂(1, 2) −
h
= Ŵ(2, 1) ,
(A.7)
where ρ 0 (1) = Ψ† (1)Ψ(1) − hΨ† (1)Ψ(1)i, v̂(1, 2) = v̂(x1 , x2 )δ(t1 −
t2 ). Ŵ(1, 2) essentially gives the potential at point 1 due to the
presence of a test charge at point 2, including the effect of the polarization of electrons. Ŵ represents the effective interaction between two electrons, and is much weaker than the bare Coulomb
94
ABOUT CORRELATIONS
interaction v̂ if the polarizability is large. The first two terms in
the expansion of Σ̂ are
Z
Σ̂(1, 2) = ihG(1, 2)Ŵ(1+ , 2) − h2 d(3)d(4) G(1, 3)
×
G(3, 4)G(4, 2)Ŵ(1, 4)Ŵ(3, 2) + · · ·
,
(A.8)
with 1+ = x1 , t1 + ∆. The first terms in the expansion of Σ̂ are represented by the Feynman’s diagrams reported in Figure 35. There
are just one first- and second-order and six third-order terms, being the order given by the number of wiggly lines in the diagram.
If the screened potential Ŵ is written in terms of Green’s functions instead of density-density correlations [104], equation A.5
becomes solvable self-consistently. The practical usefulness of
this scheme depends on how many terms in the expansion are
needed to provide a good approximation.
95
B
T H E PA W M E T H O D
The numerical difficulties in solving the Kohn-Sham equation
1.8 come from the very different behavior of the wavefunction
in different regions in space. In the atomic region around the nucleus, the wavefunctions present rapid oscillations, which require
fine grids for accurate numeric calculations. However, the wavefunctions can be expressed by a small basis set through proper
choices, such as (the one consisting of) atomic-like orbitals. In the
bonding region between the atoms, on the contrary, the wavefunctions have smooth variations. Nevertheless, they are very susceptible to changes in the environment, requiring for this reason
large basis sets. This is the source of the difficulty of electronic
structure methods to describe the bonding region to a high degree of accuracy while accounting for the large variations in the
atom center.
Many attempts have been done to overcome this drawback. For
example, the strategy of the augmented-wave methods is to divide the space into atom-center spherical regions inside which
wavefunctions are expanded in so-called partial waves, mainly
atomic-like wavefunctions, and interstitial bonding regions outside the spheres where wavefunctions are expanded in envelope
functions, usually plane waves. Envelope functions and partial
waves expansions are then matched at the boundary between
the atomic and interstitial regions. The projector AugmentedWave method (PAW) [106] is an all-electron method inspired from
the augmented-wave one, but it approaches the problem in a
slightly different way. Starting from the above observation that
valence all-electron (AE) Kohn-Sham wavefunctions (orthogonal
to the core states) exhibits strong oscillations near the nuclear
positions, the PAW method searches a linear transformation that
maps these wavefunctions into new, so-called pseudo (PS), computationally convenient wavefunctions (remind that an AE wavefunction is a full one-electron Kohn-Sham orbital and is not to
be confused with a many-electron wavefunction). The PS wave-
96
THE PAW METHOD
functions will be identified with the enveloped functions of the
augmented-wave method. In this section, all quantities related to
PS representation will be indicated by a tilde; in this way we may
write that
Ψ = T̂ Ψ̃ ,
(B.1)
where for sake of clearness Ψ and Ψ̃ are respectively the AE
and PS wavefunctions and T̂ is the linear transformation operator which maps the AE Hilbert space into the PS one. As the
PS wavefunctions are intended to avoid the complicated nodal
structures around the nuclei, the linear transformation T̂ is constructed as a sum of local atom-centered transformations T̂R that
act only within some augmentation region ΩR (R refers to atomic
sites) enclosing the atoms; thus
X
T̂ = 1 +
T̂R ,
(B.2)
R
so that, by construction, the AE and PS wavefunctions coincide
outside the augmentation regions, as illustrated in Figure 36.
The local terms T̂R are defined for each augmented region individually by specifying the target functions |φi i of the transformation T̂ for a set of initial functions |φ̃i i that is orthogonal to
the core states and complete in the augmentation region, namely
"
#
X
|φi i = 1 +
T̂R |φ̃i i .
(B.3)
R
Initial states |φ̃i i and target functions |φi i are called PS partial
waves and AE partial waves respectively. A natural choice for
the AE partial waves are the solutions of the radial Schrödinger
equation for the isolated atom, so that the index i refers to the
atomic site R, the angular momentum quantum numbers L =
(l, m), and an additional index n to label different partial waves
for the same site and angular momentum. The PS partial waves
must be identical to the corresponding AE partial waves outside
the augmentations regions and should form a complete set of
functions within. For these reasons, every PS wavefunction can
be expanded into PS partial waves in ΩR :
X
|Ψ̃i =
|φ̃i ici .
(B.4)
i
97
THE PAW METHOD
Since |φi i = T̂ |φ̃i i, the corresponding AE wave function is in ΩR
of the form
X
|Ψi = T̂ |Ψ̃i =
|φi ici ,
(B.5)
i
with identical ci coefficients as in B.4.
Figure 36: Schematic illustration of an atomic AE wavefunction |Ψi (red
line) and the corresponding atomic PS wavefunction |Ψ̃i (blue
line), as well as the Coulomb potential (orange line) and pseudopotential (green line). (From [107]).
Therefore, the AE wavefunctions can be written as
X
X
|Ψi = |Ψ̃i −
|φ̃i ici +
|φi ici ,
i
(B.6)
i
where the expansion coefficients for the partial wave expansions
remain to be determined.
98
THE PAW METHOD
As T̂ is a linear operator, the coefficients ci are defined as the
scalar products ci = hp̃i |Ψ̃i, where
hp̃i | is a projector function
P
which fulfills the completeness i |φ̃i ihp̃i | = 1 and orthogonality
hp̃i |φ̃j i = δij relations. In this way, equation B.6 can be rewritten
in the usual form
X
|Ψi = |Ψ̃i +
(|φi i − |φ̃i i)hp̃i |Ψ̃i .
(B.7)
i
As evident, three quantities determine this transformation: (a)
the AE partial waves |φi i which are obtained by radially integrating the Schrödinger equation for the isolated atom; (b) the PS partial waves |φ̃i i which coincide with the corresponding AE partial
waves outside the augmentation regions; (c) one projector function |p̃i i for each PS partial wave localized within the augmentation region. The partial waves are thus functions on a radial grid,
multiplied with spherical harmonics. The PS wavefunctions are
usually expanded into plane waves, but other choices are equally
possible. The projectors are also calculated as a radial function
times the spherical harmonics.
Physical quantities of interest are obtained from the expectation values of the corresponding P
operators. Given an operator
A, its expectation value is hAi = n fn hΨn |A|Ψn i, where fn is
the occupation number
P of the state n. Alternatively, it can be
calculated as hAi = n fn hΨ̃n |Ã|Ψ̃n i, where à = T̂ † AT̂ acts on
the pseudo wavefunctions; for a quasi-local operator A, the PS
operator à assumes the expression
X
à = A +
|p̃i i hφi |A|φj i − hφ̃i |A|φ̃j i hp̃j | .
(B.8)
ij
P
The adding of a term as B − ij |p̃i ihφ̃i |B|φ̃j ihp̃j |, where B is
located within the augmentation region, does not change the expectation value of the PS operator Ã, but avoids the singularity of
the Coulomb potential at the nuclear site, as shown in Figure 36,
and leads to an expression that is less sensitive to the truncation
of plane waves’ number.
The charge density at a point r in space is given by the expectation value of the projection operator |rihr|, obtaining from B.8
n(r) = ñ(r) + n1 (r) − ñ1 (r)
,
(B.9)
99
THE PAW METHOD
where
ñ(r) =
X
fn hΨ̃n |rihr|Ψ̃n i
n
n1 (r) =
X
fn hΨ̃n |p̃i ihφi |rihr|φj ihp̃j |Ψ̃n i
(B.10)
n,(ij)
ñ1 (r) =
X
fn hΨ̃n |p̃i ihφ̃i |rihr|φ̃j ihp̃j |Ψ̃n i
.
n,(ij)
n1 (r)
The terms
and ñ1 (r) are localized around each atoms and
can be calculated in spherical coordinates. Their notation is faithful with the original work of P.E. Blöchl [106].
Figure 37: (left): Comparison of the PS wavefunctions (solid lines) and
the corresponding AE wavefunctions (dashed lines) for the
configurations 3s2 3p2 and 3s1 3p2 3d1 of silicon. (right): Nonlocal PS potential of Si for the angular momentum l = 0, 1, 2.
The dashed line denotes the Coulomb potential of a point-like
atomic core. (From [108]).
For completeness, in Figure 37 we report as example the AE
wavefunctions 3s 3p and 3d for the silicon atom, and the corresponding nodeless PS wavefunctions; the latter coincide with the
true wavefunctions for large r, and appropriately extrapolate to
zero for small r. The non-local PS potentials for l = 0, 1, 2 are also
100
THE PAW METHOD
reported, and is evident how they are rather smooth compared
with the Coulomb potential of a point-like atomic core.
The PAW method is exact within the DFT framework, provided
the plane wave expansion is complete and the partial wave expansion B.4 is converged. Typically, one or two partial waves per
angular momentum (l, m) and site are used.
101
C
A PRACTICAL EXAMPLE OF
THE USE OF SYMMETRY TOOLS
In order to better understand how the symmetry tools PSEUDO
and AMPLIMODES effectively work, let us consider the practical
example of the orthorhombic structure of the organic ferroelectric CBDC. Experimentally we know that CBDC belongs to the
polar Cc space group; all the atomic positions RH
i can be written
as RG
+
u
,
where
H
and
G
refer
to
the
low-symmetry
Cc space
i
i
group and its pseudosymmetry respectively. We find the highsymmetry space group G and the distortion field ui by using the
PSEUDO tool. The starting structure can be input via the Crystallographic Information File (CIF), which is the standard format
for crystallographic data. The pseudosymmetry can be searched
by PSEUDO in different ways; the most simple procedure uses,
however, the option Minimal supergroups. Before running the tool,
a tolerance threshold must be assigned; standard values are between 1 or 2 Å, because larger thresholds can lead to unphysical
results. A list of minimal supergroups of Cc will appear on the
screen. By definition, a minimal supergroup G of the given space
group H has no subgroup that is also a supergroup of H. An
example of such a list is reported in Figure 38. Very useful informations are the so called k-index, i.e. the multiplicative factor
which relates the primitive unit cell of the given structure with
that of the hypothetical minimal supergroup, and the transformation relating both cells. From the list of minimal supergroups,
many cases can be discarded a priori by physical intuition. For
example, the number of molecules in the given structure must be
multiple of the k-index, so that the number of molecules in the supergroup primitive unit cell is an integer. The program then lists
the supergroups that have been checked and the ones for which
pseudosymmetry has been detected under the required tolerance.
For CBDC, this list is reported in Figure 39. Among the minimal
supergroups below the tolerance threshold, we choose that with
102
A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
Figure 38: List of some minimal supergroups for the space group Cc.
smaller maximum atomic displacements, reported in the last column of the list under the label umax . In our case, the space group
C2/c represents the searched pseudosymmetry, and the relative
displacement ui for each atom is shown in a following table (Figure 40). The symmetrized structure is given first in the subgroup
Cc setting (i.e the same setting of the input structure), but also in
the supergroup C2/c setting, which will be the high-symmetry
input of the AMPLIMODES tool for the symmetry mode analysis on the distortion field.
This second available tool, first of all, internally transforms the
high-symmetry parent structure into the setting of the low symmetry phase. This structure is the so-called REFERENCE STRUCTURE, because it will be used as the reference configuration to describe the atomic displacements which lead to the low-symmetry
state. In general, the number of symmetry independent atoms
in the low- and high-symmetry settings is different. The input
high-symmetry structure as calculated by PSEUDO and the REFERENCE STRUCTURE are listed in tables reported in Figures
41 and 42 respectively. In the present case, the number of independent atoms increase from 9 to 16, because by symmetry all
103
A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
Figure 39: List of minimal supergroups below the required tolerance.
8f Wyckoff positions split into two 4a WP. It is now important
to note that the cell parameters for both settings are the same.
This is just a special situation, because the transformation that
links the two settings generally is different from the unitary one.
Once the reference structure has been calculated, the program
performs an atom mapping or pairing identifying the atoms in the
low-symmetry structure which correspond to those listed in the
reference one. From the comparison of these atomic pairs, the
atomic displacements are calculated, and listed in a proper table
like that in Figure 43. The displacements reported in the above
lists for the atoms in the reference structure uniquely define the
displacive distortion relating the two structures. The Cc structure is polar along the x and z directions, and this means that
there is always an arbitrariness in the choice of the origin along
those directions. In such a situation, AMPLIMODES asks if the
user wants to perform a global translation of the structure as
a whole in order to minimize the total amplitude of the distortion. At this point, the program computes the basis of symmetry-
104
A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
Figure 40: Displacement field which relates the polar space group Cc to
its pseudosymmetry C2/c.
Figure 41: High-symmetry structure as calculated by PSEUDO.
adapted modes to describe the displacive distortion, performs
the mode decomposition of the distortion with this basis, and
prints on the screen the list of results in Figure 44. The basis of
105
A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
Figure 42: REFERENCE STRUCTURE, i.e. the high-symmetry parent
structure transformed into the setting of the low-symmetry
phase.
Figure 43: Table listing the atom mapping.
symmetry-adapted modes are chosen such that each displacive
mode acts on a single Wyckoff position of the high-symmetry
106
A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
Figure 44: Summary of symmetry modes which act on the different
Wyckoff positions of the high-symmetry structure.
structure; thereby, the total distortion can be divided into different contributions coming from different Wyckoff positions of the
parent structure. For CBDC, the pseudosymmetry C2/c has three
WP orbits (namely 8f, 4e and 4b), and consequently the distortion
consists of three contributions. The first table lists the number of
basis modes and their irreducible representations; for each Wyckoff orbit, the relevant atoms and the relevant irreducible representations, with the number of their basis symmetry modes reported
in parenthesis, are listed. This information depends exclusively
on the WP type, therefore all atoms with the same type of orbit
are included in a single row. In our example, there are 48 basis
modes, 22 corresponding to the irreducible representation GM1+
and 26 for GM2-. In fact, let us consider as example the WP 8f; we
need for each one of the seven atom three basis modes for each irreducible representations. The second list reports the amplitude
of the distortion associated with each irreducible representation,
107
A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS
and its isotropy subgroup that gives informations about symmetry properties. For CBDC, we can see that the amplitude of distortion associated to GM1+ is zero, whereas on the other hand the
global distortion associated to GM2+ is 1.65 Å. From the isotropy
group of this latter irreducible representation, which is polar, we
deduce that the distortion can induce a non-zero dipole moment
in the structure.
108
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