Density Functional Theory and Group Theoretical Analysis in the Study of Hydrogen Bonded Organic Ferroelectrics

Transcription

1 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

2 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.

3 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.

4 A C K N O W L E D G E M E N T S 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

5 A B S T R A C T 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...

6 C O N T E N T S Introduction 1 I Theoretical and Computational Methods 5 1 THE DENSITY FUNCTIONAL THEORY The Many-Body Problem The Hohenberg-Kohn Theorem The Kohn-Sham Equations The Exchange-Correlation Term Exact Properties for E xc [n] Local Spin Density Approximation 16 2 DFT CALCULATIONS PBE functional Hybrid functionals The Electronic Ground State Optimization of Atomic Positions The Modern Theory of Polarization The VASP code 33 3 SYMMETRY ANALYSIS The Pseudo Tool The Amplimodes Tool 36 II Computational Results and Analysis 38 4 POLAR DISTORTIONS IN H-BONDED ORGANIC FERROELECTRICS Structural Properties The Ferroelectric Polarization Symmetry-Mode Analysis of Ferroelectricity Conclusions 60 vi

7 CONTENTS vii 5 POST-DFT STUDY OF CROCONIC ACID PROPERTIES Structural Properties Electronic Properties 66 6 MULTIFERROICITY IN A MANGANESE BASED MOF Different types of multiferroics Metal-organic frameworks Crystal Structure and Spin Ordering of Mn-MOF Microscopic Origin of the Spontaneous Polarization Electronic Properties 83 7 CONCLUSIONS 86 III Appendix 90 a ABOUT CORRELATIONS 91 b THE PAW METHOD 96 c A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS 102 BIBLIOGRAPHY 109

8 I N T R O D U C T I O N 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 T c for the paraelectric-ferroelectric phase transition: as the temperature approaches T c, 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 T c 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

9 CONTENTS 2 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 T c 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

10 CONTENTS 3 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 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.

11 CONTENTS 4 Results regarding the organic ferroelectrics CBDC and Ph- MDA (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 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.

12 Part I Theoretical and Computational Methods 5

13 1 T H E D E N S I T Y F U N C T I O N A L T H E O R Y 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

14 1.1 THE MANY-BODY PROBLEM 7 obtained about the nuclei and electrons in the system. The full Hamiltonian can be expressed as Ĥ = ˆT e + ˆT n + ˆV ee + ˆV nn + ˆV en = h 2 2 i 2m h 2 2 I e 2M i I I + 1 e 2 2 r i j i r j + 1 Z I Z J e 2 2 R I J I R J Z I e 2, (1.2) r ii i R I where ˆT e and ˆT n refer to kinetic energies of electrons and nuclei respectively, and ˆV ee, ˆV nn, ˆV en 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 Ĥ e (r; R) given by (r and R are shorthand notations) Ĥ e (r; R) = ˆT e + ˆV ee + ˆV nn + ˆV en = ˆT e + ˆV(r; R). (1.3) In this Hamiltonian the variables R appear simply as parameters (instead of quantum dynamical observables); thus Ĥ e (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 Ĥ e (r; R) is Ĥ e (r; R)Ψ n (r; R) = E n (R)Ψ n (r; R), (1.4) where the electronic wavefunctions Ψ n (r; R) and the eigenvalues E n (R) depend on the parameters R; the suffix n summarizes the electronic quantum numbers. Even within the Born- Oppenheimer 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.

15 1.2 THE HOHENBERG-KOHN THEOREM 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 ˆV en 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 V ˆ en (such that ˆV en V ˆ en cost) eigenvalues equation 1.4 gives a ground state wavefunction Ψ 0 (r; R), and suppose for absurd that n[ ˆV] = n[ ˆV ]; clearly the following relations must hold: E 0 = Ψ 0 Ĥ e Ψ 0 = Ψ 0 Ĥ e + ˆV en ˆV en Ψ 0 < Ψ 0 Ĥ e + ˆV en ˆV en Ψ 0 = E 0 + dr n(r)[ ˆV en(r) ˆV en (r)] ; (1.5) by reversing the primed and unprimed quantities, one obtains an absurd result, because the inequality in 1.5 is strict being Ψ 0 and Ψ 0 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: E[n] = dr n(r) ˆV en (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

16 1.3 THE KOHN-SHAM EQUATIONS 9 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 ˆV en (r) with an effective one-electron potential ˆV eff (r) holding the same ground state charge density. The total energy functional 1.6 can be rewritten as E[n] = dr n(r) ˆV en (r) + T s [n] + E xc [n] + e2 2 drdr n(r)n(r ) r r, (1.7) where the functional T s [n] is the kinetic energy of the non interacting system and the last term, E Hartree [n], represents the classical Coulomb interaction energy of an electronic cloud of density n(r). The term E xc [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 [ h 2 ] 2 + ˆV eff (r) Ψ i (r) = ε i Ψ i (r), (1.8) 2m e where ε i and Ψ i (r) are the so called Kohn-Sham eigenvalues and eigenfunctions respectively. ˆV eff (r) is the effective one-electron potential associated to the same ground state charge density of the interacting many-particle potential: ˆV eff (r) = ˆV en (r) + e 2 dr n(r ) r r + ˆV xc (r), (1.9) being the exchange-correlation potential given by the functional derivative ˆV xc (r) = E xc[n] n(r). (1.10)

17 1.3 THE KOHN-SHAM EQUATIONS 10 In terms of Kohn-Sham orbitals, the electron charge density can be written as occ n(r) = Ψ i (r) 2, (1.11) i where the summation runs over all occupied one-electron Kohn- Sham 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 occ E[n] = ε i E Hartree [n] dr n(r) ˆV xc (r) i +E xc [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 ˆV xc (r); any difficulty in the solution procedure has been confined to the choice of a reasonable guess of the exchange-correlation functional E xc [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 ε i don t have any physical meaning, since the Koopmans theorem doesn t hold for them. The identification of ε i 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

18 1.4 THE EXCHANGE-CORRELATION TERM 11 functional appears, while in the Hartree-Fock equations the exchange term is taken into account exactly: ˆV x HF (r)ψ i (r) = e 2 δ si s j dr Ψ j (r)ψ j (r ) r r Ψ i (r ) (1.13) 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 E xc [n] with a function of the local density n(r) writing E LDA xc [n] = dr n(r)ε xc (n(r)), (1.14) where ε xc (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 ε xc = /r s /( r s r s ) = /r s ln r s r s r s ln r s (1.15)

19 1.4 THE EXCHANGE-CORRELATION TERM 12 respectively for r s 1 and r s 1, where r s = (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: ε x = 3 4 e2 ( 3 π) 1/3 [n(r)] 1/3. (1.16) Other parameterizations appeared in literature yield similar results, since they are very similar in the range of r s 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 E xc = 1 2 drdr n(r) n xc(r, r ) r r, (1.17) where n xc (r, r ) 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 ) giving the probability to find an electron in r if there is already one in r, the exchange-correlation hole is defined as n xc (r, r ) = n(r ) 1 0 dλ[g λ (r, r ) 1], (1.18) where g λ (r, r ) is the pair correlation function for a system in which the electron-electron interaction ˆV ee 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 n x (r, r ) 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.

20 1.4 THE EXCHANGE-CORRELATION TERM 13 Figure 2: Left: exchange hole n x (r, r ) 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 n x (r, r ) times r 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 n xc (r, r ), 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

21 1.5 EXACT PROPERTIES FOR E xc [n] 14 the problems of LDA can be avoid introducing gradient corrections, writing the exchange-correlation functional as function of the local density and its gradient: E GGA xc [n] = dr n(r)ε xc (n(r), n(r) ) ; (1.19) 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 E c [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 E xc [n] is typically much smaller than the other known terms T s [n], E Hartree [n] and ˆV en, with the hope that a reasonable simple guess for E xc [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] E x [n λ ] = λe x [n], E c [n λ ] > λe c [n] for λ > 1 E c [n λ ] < λe c [n] for λ < 1 (1.20)

22 1.5 EXACT PROPERTIES FOR E xc [n] 15 where n λ = λ 3 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 E c [n (1) ] = 0, E x [n (1) ] = E Hartree [n (1) ] (1.21) where n (1) 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]: E x [n] E xc ][n] 1.68e 2 dr n(r) 4/3 ; (1.22) LDA and many (but not all) GGAs satisfy An important feature of the E xc [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] = ˆV xc + ˆV xc = xc (1.23) δn δn N+δ 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 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].

23 1.6 LOCAL SPIN DENSITY APPROXIMATION 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 ˆV eff (r) in equation 1.8 by the spin dependent effective single-particle potential ˆV σ eff (r) = ˆV en (r) + e 2 dr n(r ) r r + ˆV σ xc (r), (1.25) where the charge density n(r) is intended as the sum of spin densities n (r) and n (r), with occ n σ (r) = Ψ i,σ (r) 2, (1.26) i,σ and Ψ i,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 ˆV σ eff (r) we take the external potential ˆV en (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 ε xc (n, n ) is the exchange-correlation energy per particle of such a system, the spin dependent exchange-correlation potential is given by ˆV σ xc d (r) = dn σ (r) [n(r)ε xc(n, n )], (1.27)

24 1.6 LOCAL SPIN DENSITY APPROXIMATION 17 so that the spin dependent exchange-correlation energy functional of the system can be written in a similar way as equation 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] ˆV xc σ (r) = r s [ β(r s ) ± 1 ] δ(r s )ζ(r) 3 1 ± 0.297ζ(r), (1.28) where ± refer to spin and respectively, and functions β(r s ) and δ(r s ) are parameterized in terms of the r s as follow: [ β(r s ) = r s ln ] r s r s δ(r s ) = r s (1.29) r s During the course of years, there have been other parameterized forms proposed for ˆV σ xc (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] E x [n, n ] = E x[2n ] + E x [2n ] 2 and the same Lieb-Oxford bound 1.22 holds., (1.30)

25 2 D F T C A L C 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

26 2.1 PBE FUNCTIONAL 19 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 E xc = E x + E c 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 E GGA xc [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 r s ) as for ε xc. 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] E GEA xc [n, n ] = E LSD xc [n, n ] + dr C σ,σ xc (n, n ) σ,σ n σ n σ /n 2/3 σ n 2/3 σ, (2.2) and then cutting off its spurious long-range parts to satisfy sum rules on the exact hole. Langreth and Perdew [35] showed that the GEA hole violates the sum rule du n c (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.

27 2.1 PBE FUNCTIONAL 20 Figure 3: Spherically averaged exchange hole density n x for s = n /2k F 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 = 2k F 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

28 2.1 PBE FUNCTIONAL 21 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 E PBE c [n, n ] = dr n[ε unif c (r s, ζ) + H(r s, ζ, t)], (2.3) where ε unif c is the correlation functional for the homogeneous electron gas, r s is the known local Seitz radius (n = 3/4πr 3 s = k 3 F /3π2 ), ζ = (n n )/n is the relative spin polarization, and t = n /2φk s n is a dimensionless density gradient. Here φ(ζ) = [(1 + ζ) 2/3 + (1 ζ) 2/3 ]/2 is a spin-scaling factor, and k s = 4k F π is the Thomas-Fermi screening wave number (a 0 = h 2 /me 2 ). The term H in 2.3 can be written as H = e2 γφ 3 ln {1 + βγ [ 1 + At 2 ]} a t At 2 + A 2 t 4, (2.4) where A = β [exp { εunif c /(γφ 3 e 2 /a 0 )} 1] 1 (2.5) γ and γ β 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 (e 2 /a 0 )βφ 3 t 2 ; (b) in the opposite t limit H, making correlation vanish; (c) under uniform scaling to the high-density limit [n(r) λ 3 n(λr) and λ ] the correlation energy must scale to a constant, thus H (e 2 /a 0 )γφ 3 ln t 2 in order to cancel the logarithmic singularity of ε unif ε unif c c. The exchange term in PBE functional is constructed satisfying four further conditions: (d) under the same uniform density scaling of condition (c), E x must scale linearly as function of λ (see the first relation in 1.20), so that E PBE x [n, n ] = dr nε unif x (n)f x (s), (2.6) where s = n /2k F n is another dimensionless density gradient, and ε unif x given by 1.16; (e) the exact exchange energy must

29 2.1 PBE FUNCTIONAL 22 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 F x (s) 1 + µs 2 with µ ; (g) the Lied-Oxford bound 1.22 is verified only if F x (s) A simple enhancement exchange factor F x (s) which satisfies all these conditions is F x (s) = 1 + κ κ/(1 + µs 2 /κ), (2.7) where κ = 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]).

30 2.2 HYBRID FUNCTIONALS 23 E PBE xc [n, n ] = dr nε unif x (n)f xc (r s, ζ, s), (2.8) where F xc (r s, ζ, s) defines the enhancement exchange-correlation factor [16, 29]. LSD can be seen as a further approximation, replacing F xc (r s, ζ, s) in 2.8 with its zero-gradient value F xc (r s, 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 s 3 and 0 r s 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 (r s >> 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 E xc [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 Ψ 0 of one-electron Kohn-Sham orbitals which yields those densities and minimize the expectation values of the kinetic energy operator ˆT and of the exact Kohn-Sham exchange energy E x = Ψ 0 ˆV ee Ψ 0 e2 2 drdr n(r)n(r ) r r, (2.9) where ˆV ee is the electron-electron repulsion operator and n = n + n. Hybrid functionals which incorporate some of this ex-

31 2.2 HYBRID FUNCTIONALS 24 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 where E xc = 1 0 dλe xc,λ, (2.10) E xc,λ = Ψ λ ˆV ee Ψ λ e2 2 drdr n(r)n(r ) r r, (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 E xc,λ=0 = E x. Assuming that at the end-point (λ = 1) E xc,λ=1 = E DFT xc, the most simple hybrid functional which approximates equation 2.10 is E hyb xc = 1 2 (E x + E DFT xc ), (2.12) where for E DFT xc it is possible to assume every local or semilocal 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 E DFT xc has a couplingconstant decomposition kernel E DFT xc,λ like equation 2.10, which permits to define E DFT x = E DFT xc,λ=0 and EDFT c = E DFT xc E DFT x. Perdew Ernzerhof and Burke [41] proposed the following simple model for the hybrid coupling-constant dependence: E hyb xc,λ (n) = EDFT xc,λ + (E x E DFT x )(1 λ) n 1, (2.13)

32 2.2 HYBRID FUNCTIONALS 25 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 E hyb xc = 1 0 dλe hyb xc,λ = EDFT xc + 1 n (E x E DFT x ), (2.14) 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): E hyb xc = E GGA xc (E x E GGA x ). (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/k 2 of the 1/r Coulomb potential for k = 0. This singularity can

33 2.2 HYBRID FUNCTIONALS 26 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) = SR + LR = + erf(ωr) 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 E PBE0 xc = ae x + (1 a)e PBE x + E PBE c, (2.17) where the exchange term is given by E PBE0 x = ae x + (1 a)e PBE x, and a = 1/4; the splitting into short- and long-range components leads to E PBE0 x = ae SR x (ω) + ae LR x (ω) + (1 a)ex PBE,SR (ω) +E PBE,LR x (ω) aex PBE,LR (ω). (2.18) Numerical tests based on realistic ω values (for example ω = 0.15) indicate that the long-range exchange contributions ae LR x (ω) and ae PBE,LR x (ω) of this functional are rather small and tend to cancel each other. Neglecting these two terms in 2.18, the HSE hybrid functional is written as xc = ae SR x (ω) + (1 a)e PBE,SR x (ω) (ω) + E PBE c (ω). (2.19) E HSE +E PBE,LR x 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 E B3 xc = E LSD xc +a c (E GGA c + a 0 (E x E LSD x ) + a x (E GGA x E LSD x ) E LSD c ), (2.20)

34 2.3 THE ELECTRONIC GROUND STATE 27 usually known as B3, where the parameters a 0 = 0.20 a x = 0.72 and a c = 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 E nl c [n] of the form E nl c [n] = drdr n(r)φ(r, r )n(r ), (2.21) as implemented in the vdw-df hybrid functional [44, 45]. The kernel φ is given as a function of r r f(r) and r r f(r ), with f(r ) 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

35 2.3 THE ELECTRONIC GROUND STATE 28 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 n in (r) to the Kohn-Sham equations 1.8, an unique operator  is defined such that n out (r) = Â[n in (r)] ; (2.22) at self-consistency, n(r) = Â[n(r)] must hold. In this way, the simplest algorithm implies the use of n out as the new initial guess for the charge density, e.g. n (i+1) in = n (i) out, (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 δn out is smaller than the error in input δn in. For the scheme in 2.23 the propagation of output uncertainties is given by a relation as δn out = Jδn in, (2.24) and depending on the size of the largest eigenvalue e J of the matrix J with respect to the unity, the algorithm could converge or not. Usually e J > 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 n (i+1) in = (1 α)n (i) in + αn(i) out, (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/e J. 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

36 2.4 OPTIMIZATION OF ATOMIC POSITIONS 29 waves), the Kohn-Sham equations 1.8 take the form of a secular equation: G H(k + G, k + G )Ψ k,i (G ) = ε k,i Ψ k,i (G ), (2.26) where H(k + G, k + G ) 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 N pw N pw Hermitian matrix, where N pw 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 N pw N pw matrix grows as N 3 pw, and the storing requires a computer memory which scales as N 2 pw. 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 E = E(Ψ k,i (G)) [ ] λ ij Ψ k,i (G)Ψ k,j(g) δ ij ij G, (2.27) with respect to the variables Ψ k,i (G) and the Lagrange multipliers λ ij. In general, the gradient of E 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

37 2.4 OPTIMIZATION OF ATOMIC POSITIONS 30 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 E tot [n] = E[n] + E nn, (2.28) where the term E nn refers to the Coulomb ion-ion repulsion energy. We can write the force acting on the atom in the position R i as F i = Ri E tot = dr n(r) Ri V en (r) Ri E nn F i, (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] F i = dr [ Ri Ψ k (r)(ĥ KS ε k )Ψ k (r) k + Ri Ψ k (r)(ĥ KS ε k )Ψ k (r)], (2.30) which clearly vanishes if Ψ k (r) and ε k are the ground-state eigenfunctions and eigenvalues of the Kohn-Sham Hamiltonian Ĥ KS. In this way, only the expectation value of the Coulomb electronion interaction gradient Ri V en (r) and the Coulomb ion-ion repulsion energy gradient Ri E nn give rise to the force acting on the i-th atom. Unfortunately, the term F i in 2.30 vanishes only if the groundstate charge density n(r) and wavefunctions Ψ k (r) are perfectly converged, and this is never the case in real calculations because

38 2.5 THE MODERN THEORY OF POLARIZATION 31 wavefunctions are expanded on a finite size basis set which cannot be complete. Nonzero F i 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 R i. 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 P = P ion + P el = 1 NΩ j z j er j e NΩ dr n(r)r, (2.31) where e is the absolute value of the electronic charge and R j are the positions within the crystal of the nuclei of charge z j 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

39 2.5 THE MODERN THEORY OF POLARIZATION 32 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 Ψ n (k, r, λ) indicate the Kohn- Sham 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 Ψ n (k, r, λ), we can express the electronic contribution to the average crystal polarization in the form P el (λ) = e dr n(r)r NΩ NΩ = 2e Ψ n (k, r, λ) r Ψ n (k, r, λ), (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 u n (k, r, λ) the periodic part of the Bloch functions, equation 2.32 can be written as P el (λ) = 2e NΩ u n (k, r, λ) r u n (k, r, λ). (2.33) nk As we noted before, only changes in polarization have real physical meaning, so we are interested in variations of P el (λ) with respect to λ: P el (λ) λ = 2e NΩ 2Re u n (k, r, λ) r λ u n(k, r, λ) (2.34) nk where Re stands for the real part. After some manipulations, equation 2.33 can be recast in the usual form P el λ = 2e 2Im NΩ k u n (k, r, λ) λ u n(k, r, λ) ; (2.35) nk the total change P el in polarization is obtained by integrating 2.34 in dλ within the range 0 λ 1 and in the Brillouin zone.

40 2.6 THE VASP CODE 33 In practice, integration over the three-dimensional Brillouin zone is carried out performing integrations over one variable, say for example k z, once a number of special points are chosen for the other two variables k x and k y. From these assumptions, the final form that one can get for the n-th band contribution, after a little of algebra, is P el = e πab γ n(c) (2.36) = e πab Im u n (k z, r, λ) kz,λu n (k z, r, λ) dl C where γ n (C) is the Berry phase of the cell-periodic wavefunctions moving along the circuit C identified by the rectangle π/c k z π/c and 0 λ 1 in the (k z, 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 P α, (2.37) e u i,β 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

41 2.6 THE VASP CODE 34 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].

42 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 R i can be described as R 0 i + u i, where u i are small displacements and where the virtual structure with atomic positions R 0 i 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

43 3.2 THE AMPLIMODES TOOL 36 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 u i - 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, g 2,, g n } of coset representatives can be chosen so that G = H + g 2 H + + g n H. (3.1) The set {1, g 2,, g n } contains all the operations of G which are not symmetry operations for H. Denoting S as the input structure, the transformed structures g i S are calculated by the PSEUDO program and compared with the original structure S. If the displacements between S and the structures g i 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].

44 3.2 THE AMPLIMODES TOOL 37 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 u i 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].

45 Part II Computational Results and Analysis 38

46 4 P O L A R D I S TO R T I O N S I N H Y- D R O G E N B O N D E D O R G A N I C F E R - R O E L E C T R I C S 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 (NaKC 4 H 4 O 6 4H 2 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, KH 2 PO 4, 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, BaTiO 3, with a polarization as large as 27 µc/cm 2 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

47 4.1 STRUCTURAL PROPERTIES 40 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, C 6 H 6 O 4 ) [67], 2- phenylmalondialde- hyde (PhMDA, C 9 H 8 O 2 ) [68], 2-fluoro-1,3- cyclohexadione (2-FCHD, FC 6 H 7 O 2 ) [69] and 4,5-dihydroxycyclopentenetrione (croconic acid, C 5 H 2 O 5 ). The latter was the first discovered single-component organic ferroelectric exhibiting a large spontaneous polarization (as large as 21 µc/cm 2 [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

48 4.1 STRUCTURAL PROPERTIES 41 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 C 2 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 Pna2 1 ). As for CBDC and all other organic ferroelectrics treated in this work, the molecule without the hydroxyl proton can restore the C 2 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.

49 4.1 STRUCTURAL PROPERTIES 42 Figure 5: CBDC. Figure 6: PhMDA Figure 7: 2-FCHD Figure 8: Croconic acid

50 4.1 STRUCTURAL PROPERTIES 43 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 Pca2 1 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.

51 4.1 STRUCTURAL PROPERTIES 44 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

52 4.1 STRUCTURAL PROPERTIES 45 theoretical results are obtained using the semilocal density approximation (PBE) for the exchange-correlation functional. PBE HSE VdW-G Expt(50 K) A Bond lengths C(5)-O(1) C(5)-O(2) C(1)-C(5) C(1)-C(4) C(1)-C(2) O(1)-H O(1) C(4)-H(41) C(4)-H(42) H O(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)-H O(4) C(3)-H(31) C(3)-H(32) H O(4) O(2 ) O(4)O(2 ) A Estimator MAD RMS 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].

53 4.2 THE FERROELECTRIC POLARIZATION 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 P s of 0.4 and 2.8 µc/cm 2 along x- and z-axes for CBDC, 9.0 µc/cm 2 along z- axis for PhMDA and 21 µc/cm 2 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

54 4.2 THE FERROELECTRIC POLARIZATION 47 Figure 10: Schematic ball-and-stick model of CBDC (top) and Ph- MDA (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.

55 4.2 THE FERROELECTRIC POLARIZATION 48 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 Ph- MDA, 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.

56 4.2 THE FERROELECTRIC POLARIZATION 49 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

57 4.2 THE FERROELECTRIC POLARIZATION 50 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/cm 2, while for PhMDA it is along the c-axis and equal to 7.0 µc/cm 2 ; 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 2- FCHD (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

58 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 51 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, Ph- MDA 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/cm 2 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/cm SYMMETRY-MODE ANALYSIS OF FERRO- ELECTRICITY 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

59 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

60 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 53 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 P 4b = (6.6, 0, 5.5), P 4e = (0.5, 0, 1.6) and P 8f = (5.5, 0, 0.4) µc/cm 2. Their sum is (12.6, 0, 6.7) µc/cm 2, which is almost equal to the total polarization P tot = (12.7, 0, 6.6) µc/cm 2 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

61 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 54 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

62 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 55 an incipient pile up of out-of-plane charge between C 2 = O 2 and C 1 = O 3, 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 P 8f (C C) = (0.85, 0, 0.43) µc/cm 2 and P 8f (C = O) = (4.79, 0, 0.1) µc/cm 2. Again, their linearity is fulfilled because P 8f (C C) + P 8f (C = O) P 8f (C C + C = O) = (5.5, 0, 0.4)µC/cm 2, 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/cm 2 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.

63 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 56 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 2- FCHD. 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 P 4c = 10.0, P 4b = 8.8, P 4d = 0.5 and P 8e = 6.0 µc/cm 2, respectively. They

64 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 57 couple almost linearly to give a total polarization of 25.3µC/cm 2 (the global distortion gives a total polarization of 24µC/cm 2 ). 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 P 4e = (3.6, 0, 2.8), P 4a = ( 13.2, 0, 8.1) and P 8f = (3.8, 0, 4.0) µc/cm 2,

65 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 58 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.

66 4.3 SYMMETRY-MODE ANALYSIS OF FERROELECTRICITY 59 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 P 4e+4a+8f = ( 5.8, 0, 1.3) µc/cm 2, which is very different from the polarization P tot = ( 8.3, 0, 4.2) µc/cm 2 (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 P 4a+8f = ( 12.1, 0, 5.9) µc/cm 2. In this way, linearity with mode A(4e) is almost recovered, obtaining P 4e+4a+8f = ( 8.5, 0, 3.1) µc/cm 2, 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 ABO 3 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 λ 1 > λ 2 > λ 3. 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 λ i,z S = (3.4, 0.4, 0.1), and for the intramolecular hydrogen λ i,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

67 4.4 CONCLUSIONS 60 to symmetry analysis [73], infrared irreducible representations (namely A u and B u ) 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 B u, 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 λ i,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 B u. 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 λ 1,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 λ 1,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 2- FCHD, which fulfill the necessary conditions required to host ferroelectricity. We studied their ferroelectric properties through DFT-based calculations, and found a spontaneous polarization as

68 4.4 CONCLUSIONS 61 large as 14.3, 7.0, 24 and 9.3 µc/cm 2, 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 intra inter buckling tilting CBDC - PhMDA croconic FCHD - Table 2: Summary table of active polar modes for each organic ferroelectric studied.

69 4.4 CONCLUSIONS 62 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.

70 5 P O S T- D F T S T U D Y O F C R O - C O N I C A C I D P R O P E R T I E S 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 Perdew- Burke-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

71 5.1 STRUCTURAL PROPERTIES 64 LDA PBE PBE0 HSE B3LYP VdW-G X-ray neut.scat. synchr.rad. Bond Å C1-O (4) 1.285(9) (4) C2-O (4) 1.310(6) (6) C3-O (5) 1.201(6) (3) C4-O (4) 1.207(6) (6) C5-O (2) 1.257(8) (4) C1-C (5) 1.400(9) (4) C2-C (3) 1.489(7) (5) C3-C (6) 1.521(6) (5) C4-C (6) 1.481(8) (4) C5-C (6) 1.446(5) (6) O1-H (8) 0.92(2) O2-H (6) 0.849(12) Angle deg O1-C1-C (4) 128.7(3) (12) O2-C2-C (4) 121.5(3) (11) 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.

72 5.1 STRUCTURAL PROPERTIES 65 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.

73 5.2 ELECTRONIC PROPERTIES 66 LDA PBE PBE0 HSE B3LYP VdW-G MAD MAD MAD RMS RMS RMS 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-

74 5.2 ELECTRONIC PROPERTIES 67 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

75 5.2 ELECTRONIC PROPERTIES 68 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 p z 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 p z 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,

76 5.2 ELECTRONIC PROPERTIES 69 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 p z 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/cm 2 directed along the c-axis, the larger value nowadays found within the class of organic ferroelectric compounds, and in good agreement with 26 µc/cm 2 [71] calculated within the LDA approximation. Furthermore, with the VdW correction, we find a total polarization of 25 µc/cm 2, very close to LDA and PBE calculations, and

77 5.2 ELECTRONIC PROPERTIES 70 to experimental results. With the use of HSE hybrid functional, the polarization rises up to 30 µc/cm 2. 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.

78 6 M U LT I F E R R O I C I T Y I N A M A N - G A N E S E B A S E D M O F 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

79 MULTIFERROICITY IN A MANGANESE BASED MOF 72 logic (e.g., with both up and down polarization and up and down magnetization) and magnetoelectric sensors 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 BiFeO 3 and BiMnO 3, 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 YMnO 3, 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 LuFe 2 O 4 ; d) magnetic ordering, as for TbMnO 3, DyMnO 3 and TbMn 2 O 5, in which ferroelectricity is induced by magnetic long-range order, i.e. the arrangement of magnetic dipoles doesn t show inversion symmetry 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 ABX 3 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-

80 MULTIFERROICITY IN A MANGANESE BASED MOF 73 Figure 26: a) In materials like BiFeO 3 the ordering of lone pairs (yellow lobes) of Bi 3+ ions, contributes to the polarization (green arrow). b) The tilting of a rigid MnO 5 block represents the geometric frustation mechanism that generates ferroelectricity in YMnO 3. c) In charge ordered systems like LuFe 2 O 4, the coexistence of inequivalent sites with different charges, and inequivalent (long and short) bonds, leads to ferroelectricity. d) In materials like TbMnO 3, 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-

81 6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF 74 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(NH 2 ) 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 X 3 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 [CH 3 CH 2 NH 3 ]Mn[(HC OO) 3 ] with the perovskite like structure ABX 3, where the organic groups A and X are the ethylammonium [CH 3 CH 2 NH 3 ] + and the carboxylate [HCOO] respectively, and B is the divalent Mn 2+ 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 ORDER- ING OF Mn-MOF The key feature of the structure of this Mn-MOF is the anionic NaCl-framework of [Mn 2+ (HCOO) 3 ] where the nearly cubic cavities in the perovskite are occupied and the charge counterbalanced by the ethylammonium [CH 3 CH 2 NH 3 ] + cations, as in Figure 27. The compound belongs to the polar space group Pna2 1, 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 Å. We started our cal-

82 6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF 75 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 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 Å. In the ethylammonium cation, the NH 3 and CH 3 groups form both N H O and C H O weak hydrogen bonds to the oxygens atoms of the framework. The octahedrally coordinated Mn 2+ ions (d 5 with t 3 2 e2 electronic configuration) lies approximately at the center of a slightly distorted octahedron with two short (s) and two long (l) equatorial Mn O eq bond lengths (2.17, 2.18 Å and 2.21, 2.23 Å respec-

83 6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF 76 tively) and one short and long apical Mn O ap (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 HCO ap O ap groups, whereas parallel chains are linked by equatorial HCO eq O eq groups. Within the ab-plane, the Mn O eq units display an alternate pattern of elongated axes, defined by the long Mn O eq bonds, and shortened axes, defined by the short Mn O eq bonds (see left panel of Figure 28). This situation is similar to the antiferro-orbital ordering [92] on Cu sites in the well-studied KCuF 3 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 Cu 2+. 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 = 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-

84 6.1 CRYSTAL STRUCTURE AND SPIN ORDERING OF Mn-MOF 77 ations, long and short Mn O eq 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 O ap 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 4.5µ B, consistent with the d 5 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 and ev/unit cell, respectively. Mapping the system onto a Heisenberg model H = ij J ijs i S j, we find for the intra- and inter-chain coupling constants (see Figure 29) J c 4 and J a 8 cm 1 respectively, with S i S j = ±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.

85 6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION 78 Figure 29: G-type AMF spin ordering for the Mn-MOF. J c and J a refer to the exchange coupling constants of the Heisenberg model. 6.2 MICROSCOPIC ORIGIN OF THE SPONTA- NEOUS 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-

86 6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION 79 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 Pna2 1, 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 Mn- MOF. 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

87 6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION 80 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/cm 2 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

88 6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION 81 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 P 4a = (0, 0, 0.37), P 4c = (0, 0, 2.08) and P 8d = (0, 0, 3.61) µc/cm 2. Notably, their sum, P 4a+4c+8d = (0, 0, 1.90) µc/cm 2, 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 BiFeO 3 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

89 6.2 MICROSCOPIC ORIGIN OF THE SPONTANEOUS POLARIZATION 82 Figure 32: Displacement patterns (arrows) connecting centric to polar structures for atoms belonging to specified Wyckoff positions (top to bottom) for Mn-MOF.

90 6.3 ELECTRONIC PROPERTIES 83 of the perovskite, with a net dipole moment of P A = (0, 0, 2.66) µc/cm 2 in the λ = 1 phase. However, the distortion associated with the ligand groups X 3 cannot be neglected (it contributes with an opposite moment P X3 = (0, 0, 1.48) µc/cm 2 ), as well as the manganese ions that bring in a contribution P B = (0, 0, 0.37) µc/cm 2. If we take their sum, we find a polarization P A+B+X3 = (0, 0, 1.55) µc/cm 2, in good agreement with that calculated for the global polar distortion P ABX3 = (0, 0, 1.64) µc/cm 2 (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 d 5 electronic configuration of divalent Mn 2+ ions. The two spikes just below the Fermi energy can be identified with e g and t 2g orbitals, even though the lack of a perfect cubic symmetry and the octahedra tilting with respect to the z-axis produce the mixing of d xy, d xz and d yz bands with d 3z 2 r 2 and d x 2 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

91 6.3 ELECTRONIC PROPERTIES 84 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 U eff =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 d 5 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 U eff =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 U eff =4 ev, and increase the hybridization of t 2g orbitals with oxygen ones, leaving e g bands almost invariant (see bottom panel of Figure 33). For

92 6.3 ELECTRONIC PROPERTIES 85 U eff 2 ev, the t 2g 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 ABX 3 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 ABX 3 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.

93 7 C O N C L U S I O N S 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, Pna2 1, Pca2 1 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/cm 2, 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/cm 2 for CBDC, PhMDA and croconic acid (for 2-FCHD intermolecular proton transfer is strongly correlated to a buckling polar mode), with a 86

94 CONCLUSIONS 87 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 p z -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/cm 2 respectively, in agreement with the experimental result of 21 µc/cm 2 [71]. With HSE, the polarization rises up to 30 µc/cm 2, resulting from a larger total distortion amplitude of the polar mode with

95 CONCLUSIONS 88 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 ABX 3 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 [Mn 2+ (HCOO) 3 ] where the nearly cubic cavities in the perovskite are occupied and the charge counter-balanced by the ethylammonium [CH 3 CH 2 NH 3 ] + cations. We found that in the ferroelectric state, the structure presents a total dipole moment of 1.6 µc/cm 2, 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 for a divalent Mn 2+ ion. Mapping the system onto a Heisenberg model H = ij J ijs i S j, we found J c 4 and J a 8 cm 1, with S i S j = ±25/4 (+ or depending on the relative orientations of spins in the i and j sites) and J c and J a 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 U eff =U-J=4 ev produces in this case a larger hybridization of t 2g 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-

96 CONCLUSIONS 89 strated that DFT-based calculations and group theoretical analysis are very useful investigation methodologies in this fascinating field of modern materials engineering.

97 Part III Appendix 90

98 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 E c [n] 0. Since a single Slater determinant is itself more general than a simple product of indipendent one-electron orbitals, E x [n] 0 must hold. For these reasons, the upper bond E xc [n] 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]: ˆV ee = e2 2 drdr ˆn(r) ˆn(r ) ˆn(r)δ(r r ) r r, (A.1) in which the operator character is carried by the density operator ˆn(r), and the term with the delta-function subtracts out the 91

99 ABOUT CORRELATIONS 92 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 ˆn(r) = n(r) + δ ˆn fluc, one can see that the expectation value of the operator A.1 differs from the classical Coulomb interaction energy E Hartree [n] just for the contributions coming from δ ˆn fluc 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 Φ HF (r 1,, r N ) 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]: Ψ(r 1,, r N ) = exp f(r i r j ) Φ HF, (A.2) i,j where f(r i r j ) is determined by energy minimization. By means of the density fluctuation n q = i e iqr i, we can write the wavefunction A.2 in the form Ψ(r 1,, r N ) = exp q τ(q)n qn q Φ HF. (A.3) 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 Φ HF that takes into account the exchange energy, and of an exponential prefactor which has the form of independent harmonic oscillators ground-state (e x2 ), being oscillator variables proportional to n q. 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

100 ABOUT CORRELATIONS 93 Figure 35: Diagrams represented the expansion of the self-energy ˆΣ(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] G(1, 2) = i h ˆTΨ(1)Ψ (2), (A.4) where 1 stands for the five coordinates of a particle: space, spin and time, e.g. (1) = (r 1, σ 1, t 1 ) = (x 1, t 1 ); obviously the same for 2. ˆT 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) ˆV(x)]G(x, x ; ε) dx ˆΣ(x, x ; ε)g(x, x ; ε) = δ(x x ), (A.5)

101 ABOUT CORRELATIONS 94 where ĥ(x) = h 2 all nuclei 2 Z n ˆv(x, R n ) 2m e n ˆV(x) = dx ˆv(x, x )ρ(x ) ˆv(x, x ) = e2 x x ρ(x) = Ψ (x)ψ(x) = i hg(x, t; x, t + ) ( 0 + ) [ ] iε G(x, x ; ε) = d(t t ) G(x, t; x, t ) exp h (t t ). ˆΣ is the self-energy operator and takes into account the complicated correlation effects of the interacting many-particle system. A series expansion of ˆΣ in terms of the Coulomb interaction ˆv gives, as first term, the Hartree-Fock exchange potential ˆΣ HF (x, x ; ε) = ˆv(x, x ) Ψ (x )Ψ(x) = i hˆv(x, x )G(x, t; x, t + ), (A.6) which is independent of ε. However, the expansion ˆΣ in a power series in terms of ˆv 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 ˆv [104]. The potential Ŵ was first introduced by Hubbard [105] in the following way: Ŵ(1, 2) = ˆv(1, 2) i h d(3)d(4) ˆv(1, 3) ˆTρ (3)ρ (4) ˆv(3, 4) = Ŵ(2, 1), (A.7) where ρ (1) = Ψ (1)Ψ(1) Ψ (1)Ψ(1), ˆv(1, 2) = ˆv(x 1, x 2 )δ(t 1 t 2 ). 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

102 ABOUT CORRELATIONS 95 interaction ˆv if the polarizability is large. The first two terms in the expansion of ˆΣ are ˆΣ(1, 2) = i hg(1, 2)Ŵ(1 +, 2) h 2 d(3)d(4) G(1, 3) G(3, 4)G(4, 2)Ŵ(1, 4)Ŵ(3, 2) +, (A.8) with 1 + = x 1, t 1 +. The first terms in the expansion of ˆΣ 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.

103 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 Augmented- Wave 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

104 THE PAW METHOD 97 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 = ˆT Ψ, (B.1) where for sake of clearness Ψ and Ψ are respectively the AE and PS wavefunctions and ˆT 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 ˆT is constructed as a sum of local atom-centered transformations ˆT R that act only within some augmentation region Ω R (R refers to atomic sites) enclosing the atoms; thus ˆT = 1 + R ˆT R, (B.2) so that, by construction, the AE and PS wavefunctions coincide outside the augmentation regions, as illustrated in Figure 36. The local terms ˆT R are defined for each augmented region individually by specifying the target functions φ i of the transformation ˆT for a set of initial functions φ i that is orthogonal to the core states and complete in the augmentation region, namely [ φ i = 1 + ] ˆT R φ i. (B.3) R Initial states φ i and target functions φ 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 Ω R : Ψ = i φ i c i. (B.4)

105 THE PAW METHOD 98 Since φ i = ˆT φ i, the corresponding AE wave function is in Ω R of the form Ψ = ˆT Ψ = i φ i c i, (B.5) with identical c i coefficients as in B.4. Figure 36: Schematic illustration of an atomic AE wavefunction Ψ (red line) and the corresponding atomic PS wavefunction Ψ (blue line), as well as the Coulomb potential (orange line) and pseudopotential (green line). (From [107]). Therefore, the AE wavefunctions can be written as Ψ = Ψ i φ i c i + i φ i c i, (B.6) where the expansion coefficients for the partial wave expansions remain to be determined.

106 THE PAW METHOD 99 As ˆT is a linear operator, the coefficients c i are defined as the scalar products c i = p i Ψ, where p i is a projector function which fulfills the completeness i φ i p i = 1 and orthogonality p i φ j = δ ij relations. In this way, equation B.6 can be rewritten in the usual form Ψ = Ψ + i ( φ i φ i ) p i Ψ. (B.7) As evident, three quantities determine this transformation: (a) the AE partial waves φ i which are obtained by radially integrating the Schrödinger equation for the isolated atom; (b) the PS partial waves φ i which coincide with the corresponding AE partial waves outside the augmentation regions; (c) one projector function p 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 operators. Given an operator A, its expectation value is A = n f n Ψ n A Ψ n, where f n is the occupation number of the state n. Alternatively, it can be calculated as A = n f n Ψ n Ã Ψ n, where à = ˆT A ˆT acts on the pseudo wavefunctions; for a quasi-local operator A, the PS operator à assumes the expression à = A + ij p i [ φ i A φ j φ i A φ j ] p j. (B.8) The adding of a term as B ij p i φ i B φ j p j, 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 r r, obtaining from B.8 n(r) = ñ(r) + n 1 (r) ñ 1 (r), (B.9)

107 THE PAW METHOD 100 where ñ(r) = f n Ψ n r r Ψ n n n 1 (r) = f n Ψ n p i φ i r r φ j p j Ψ n n,(ij) ñ 1 (r) = f n Ψ n p i φ i r r φ j p j Ψ n. n,(ij) (B.10) The terms n 1 (r) and ñ 1 (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 3s 2 3p 2 and 3s 1 3p 2 3d 1 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

108 THE PAW METHOD 101 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.

109 C A P R A C T I C A L E X A M P L E O F T H E U S E O F S Y M M E T R Y TO O L S 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 R H i can be written as R G i + u i, where H and G refer to the low-symmetry Cc space group and its pseudosymmetry respectively. We find the highsymmetry space group G and the distortion field u i 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

110 A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS 103 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 u max. In our case, the space group C2/c represents the searched pseudosymmetry, and the relative displacement u i 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 STRUC- TURE, 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 REF- ERENCE 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

111 A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS 104 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-

112 A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS 105 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

113 A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS 106 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

114 A PRACTICAL EXAMPLE OF THE USE OF SYMMETRY TOOLS 107 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,