First Principle Study of Multiferroic Bismuth Ferrite

Examensarbete 30 hp Oktober 2018

Yuhang Liu

Masterprogrammet i fysik Master Programme in Physics Abstract

In this work, the spontaneous magnetic and electric behavior of the multiferroic R3c BiFeO3 are studied by density functional theory (DFT) with the generalized-gradient-approximation (GGA) and GGA+U method. The predicted crystal stucture is rhombohedral with space group R3c in equilibrium. The elongation of perovskitelike lattice along the [111] direction is correctly predicted by GGA+U method. We predicated a large electric polarization of 103.5 µC/cm2 by berry phase method, which is consistent with the experimental measured polarization of 100 µC/cm2 for high quality single crystal sample. Our calculations indicate that the (111) planes are magnetic easy planes and the magnetic anisotropy origins from both single-ion anisotropy and the superexchange coupling. With the superexchange included in our calculations, the ground state of BiFeO3 is found to have an approximate antiferromagnetic (AFM) order with weak ferromagnetic (FM). Our Phonon dispersion analysis shows the instability of the hypothetical FM phase of BiFeO3. We found the the energy of AFM and FM BiFeO3 evolve in opposite ways in spin-sprial dispersion and equals to each other at q=[0.34 0 -0.34]. Dedicated to all hard-working researchers at Uppsala University Contents

1 Introduction to this study ...... 7 1.1 Introduction ...... 7 1.2 Populävetenskaplig Sammanfattning ...... 11

2 Density Functional Theory ...... 13 2.1 Many Body problem ...... 13 2.2 Kohn-Sham equation ...... 14 2.3 Exchange correlation functionals ...... 15 2.3.1 Local spin density approximation (LSDA) ...... 15 2.3.2 Generalized gradient approximation (GGAs) ...... 16 2.3.3 Beyond GGA ...... 16 2.3.4 L(S)DA+U: On-site Coulomb-repulsion ...... 17 2.4 Plane wave sets and pseudopotential ...... 17 2.5 First principles phonon calculations ...... 20

3 Modern polarization theory ...... 22 3.1 Polarization lattice ...... 22 3.2 Berry phase theory for the macroscopic polarization ...... 24

4 Magnetism ...... 25 4.1 Heisenberg Hamiltonian and Magnetic ordering ...... 25 4.2 Magnetic anisotropy ...... 26 4.2.1 Single-ion anisotropy and spin-orbital coupling ...... 27 4.2.2 Non-collinear magnetism and Dzyaloshinskii-Moriya interaction ...... 27

5 Results and discussions ...... 29 5.1 Computational details ...... 29 5.2 Structure of BiFeO3 ...... 30 5.3 Electronic properties of BiFeO3 ...... 33 5.4 Electric polarization ...... 35 5.5 Magnetic anisotropy ...... 38 5.6 Spin spiral dispersion relation ...... 39

6 Summary and Outlook ...... 42

7 Acknowledgements ...... 44

References ...... 45 1. Introduction to this study

1.1 Introduction It is safe to say that the progress in science and technology is mainly focused on the understanding and application of materials. To comprehend a property of a material, the most direct and effective way is testing it by experiments. Every advancement in testing technology brings huge a progress in exploring materials, for instance, the invention of X-ray enabled researchers to study the periodic structure of crystal and the scanning tunneling microscope expanded the human’s observation to the scale of molecule and atom. However, the more advanced technology tends to consume more resource. The increasing cost and difficulty of experimental observation have made the experiments no longer the most efficient approach to study materials and even sometimes it is no longer possible. For example, to study the phase transition under ultra-high pressure, creating the high pressure in laboratory is not a wise choice and even worse, sometimes current technology cannot reach the high pressure needed. Another specific example is that people cannot and don’t need to create a star if they want to research the Sun. Various factors drive researchers to find economical and convenient replace- ment of experiment. Theory has always been a useful tool to understand the fundamentals of nature, whereas the philosophy of theory is totally different from experiment. Through an experiment, the phenomenon is always known but the origins of the phenomenon are not, and researchers obtain data from experiments and look for some theories to explain them. . If no theory can explain it, experimentalists will claim that current theories fail but a new phenomenon is found. For theory, researchers make assumptions and build up a theory based on these assumptions. If the theory agrees with experiments, it is a good theory. If the theory cannot explain what is found in experiments, theorists will claim that the theory has a flaw and needs to be improved. It is necessary to emphasize that theory can describe how the nature behaves but cannot always explain why the nature behaves like that, and theory can never be proved to be correct but can be proved to be wrong by experiments. People may wonder why we should still trust in a theory if it cannot be proved to be correct. My answer to this question is that if a theory has been proven suffi- ciently appropriate for a phenomenon during a scientific experiment then we should think that the theory will also be a suitable and credible explanation again in the future. In general, a theory is a model we build based on known phenomena through experiments and an appropriate theory is a model that

7 works every time and can be a contributing factor to our future experiments when we are explaining and analyzing unknown phenomena. At this point, I believe we should have a relatively positive and objective understanding of the theory. The quantum mechanics has been proved reliable in describing electronic system, so it should be able to calculate the properties of materials which are made up of electrons and ions. In principle, a finite or infinite electron-ion system can be described by a many-body wave function no matter how complex the system is, and if the many-body wave function is solved, all properties of this material will be reachable. However, solving such a many-body wave function for a macroscopic number of atoms is impossible for human even with the most advanced computer. For most properties, solving the many-body wave function is sufficient but not necessary. Density functional theory (DFT) transforms the many-body wave function to a single particle Kohn-Sham function and the total energy is expressed as a functional of the electron density function but not a function of the positions of all electrons[1]. Multiferroics[2][3] are materials with two or more ferroic order parameters simultaneously in the same place. The coupling between two order parameters provides novel properties and make multiferroics special. For instance, coupling between the spontaneous polarization of ferroelectricity and spontaneous magnetization of ferromagnetism results in magnetoelectric (ME) effects, in which the magnetization can be tuned by electric field and the electric polarization can be tuned by magnetic field. The magnetoelecrics with ME effects are of great interest for potential application in nondestructive data reading and writing with low energy consumption. Besides the prospective applications, the physics behind the magnetoelectric materials is still interest- ing and attractive. The magnetoelectrics are not common but actually very rare in nature. The reason is that the ions tend to be magnetic, which usually tend to the absence of electric polarization. The non-zero total magnetic moment requires partially filled 3d orbitals in transition metals or 4f orbitals in lanthanides. Whereas 3d electrons of transition metals tend to prohibit the ferroelectric distortion[4]. It has been shown in TbMnO3 that electric plar- ization can be controled by the external magnetic field[5], whereas its small polarization make it impossible to be applied in devices. The most promising material for applying ME effect in devices is BiFeO3, which is ferroelectric with a high Curie temperature of 1100 K and antiferromagnetic with a high Neél temperature of 643 K[6][7]. The structure of BiFeO3 is a highly distorted rhombohedral perovskite structure with R3c space group as shown in Fig.1.1. The distortion consists of two parts: (i) counter- rotations of neighboring O6 octahedra about the [111] direction. (ii) anions and cations displacement along [111] direction. BiFeO3 has a G-type antiferromagnetic order as shown in Fig.1.2, in which Fe magnetic moments are coupled ferromagnetically within the (111) planes and antiferromagnetically between neighbor (111) planes. In bulk BiFeO3, the magnetic order is modi-

8 fied by a spiral spin order in which antiferromagnetic vector rotates with a long wave-length of 620 Å[8]. The propagation direction is [111] in hexagonal lattice and [110] in rhombohedral lattice. The spin spiral order is suppressed in thin film BiFeO3[9]. Theoretical analysis showed that the spin spiral structure originates from the competition between the isotropic superexchange and the Dzyaloshinskii-Moriya interaction of the nearest neighbor spins of Fe atoms[10]. In thin films at room temperature, electric polarization of 50-60 µC/cm2 has been measured in experiments[11]. Theoretical studies using DFT has predicted a large polarization of 90-100 µC/cm2[12]. In bulk sam- ples, a lower polarization value of 8.9 µC/cm2 at room temperature has been measured[13][14], which is in sharp contrast with the large atomic displacements of BiFeO3. A very large spontaneous electric polarization above 100 2 µC/cm was measured in single crystals of BiFeO3 synthesized by a flux growth method[15]. The hybridization interaction between Bi-O and Fe-O has been found playing important role in the ferroelectric polarization of R3c BiFeO3 in theoretical study[16]. The purpose of this study is to calculated the electric polarization and magnetic order and to understand the origin of electric and magnetic properties of BiFeO3. At the beginning, the ground state crystal structure of BiFeO3 is predicted by DFT with GGA+U method. The most appropriate U value is determined by comparing the calculated structural parameters and experimental results. The spontaneous polarization is calculated along a assumed transition path by the berry phase method. To study the origin of magnetic order and electric polarization, the centrosymmetric R3c¯ phase of BiFeO3 is also studied by DFT and compared with the R3c BiFeO3. The more accurate magnetic order with canted magnetic moments is calculated in self-consistent calculations in which the superexchange is included. To understand the difference of the ground state AFM phase and the hypothetical FM phase of BiFeO3, the phonon dispersion and spin spiral dispersion are calculated.

9 Figure 1.1. Schematic representation of the structure of R3c rhombohedral BiFeO3

Figure 1.2. G type antiferromagnetic order

10 1.2 Populävetenskaplig Sammanfattning Framför allt kan man säga att framstegen inom vetenskap och teknik fokuserar mer påförstoch tillämpning av material. För att förståen egenskap av ett material mett direkt och effektivt experiment utföras. Varje framsteg inom provn- ingsteknik ger stora framsteg till materialforskning, till exempel röntgenteknik och kristallografi. Genom Röntgenteknik och kristallografi har forskare mer möjligheter att ta reda påden periodiska kristallstrukturen medan sveptun- nelmikroskopet expanderade människans iakttagelser till en molekylär och atomär nivå. Emellertid föreligger en tendens att de avancerade teknik kommer att konsumera mer resurser. ökade kostnader och svi experimentobserva- tion har gjort experimentet inte längre det mest effektiva sättet att studera och forska. I vissa fall är det omöjligt att utföra experiment längre. Till exempel att det inte är ett klokt val att skapa ett högtryck i laboratoriet för att studera fasövergunder ultrahögtryck och ännu värre, ibland kan den nuvarande tekniken inte nådet höga trycket som behövs. Ett annat specifikt exempel är att människor inte kan och behöver inte skapa en stjärna om de vill undersöka solen. Olika faktorer tvingar forskare att hitta en ekonomisk och praktisk ersät- tning av experiment. Teori har alltid varit ett användbart verktyg för att förstå- naturens grundprinciper men teorifilosofin är helt annorlunda än experiment. Genom ett experiment kan olika fenomen alltid observeras men fenomenens ursprung är dock svatt veta. Brukar fforskare data frett experiment och de letar efter nteorier som kan vara en lämplig förklaring till fenomenen samt sina upphov. Om det inte finns nteori kan förklara upphovsorsaker, dåhävdar experimentalister att teorier misslyckas och ett nytt fenomen hittas. För teori gör forskare antaganden och bygger upp en teori utifrdessa antaganden. Om teorin överensstämmer med experiment, dåär den en bra teori. Om teorin inte kan förklara vad som finns i experiment, kommer teoretiker att hävda att teorin föreligger en brist och behöver alltsåförbättras. Det är nödvändigt att betona att olika teorier kan beskriva hur naturen beter sig men de kan inte förklara varför naturen beter sig så. Pådet andra sätet kan teorier aldrig bevisas vara korrekta men kan motbevisas vara fel genom experiment. Människor kanske undra varför vi fortfarande sätta vlit till en teori om det inte kan bevisas vara korrekt. Mitt svar påden här frär att om en teori har bevisats tillräckligt lämplig för ett fenomen under ett vetenskapligt experiment dåbör vi tro att teorin även kommer att vara en lämplig och trobar förklaringar igen i framtiden. I generell är teorier en modell som vi har byggt utifrkända fenomen genom experiment. Lämpliga teorier är ocksåen modell som kan vara en bidragande faktor till vkommande experiment när vi ska förklara och analysera okända fenomen i framtiden. Vid det här laget tror jag att vi bör ha en relativt positiv och ob- jektiv förstav teorier. Kvanteringsmekaniken har visat sig vara tillförlitlig vid beskrivningen av elektroniskt system och därför har vi möjlighet att beräkna olika egenskaper hos material som tillverkas av elektroner och joner. I prin-

11 cip kan en Schrodinger-ekvation och alla andra partiklar i systemet ge oss alla information om det här systemet men det är bara en princip. I praktiken kommer lösningen av en smonsterekvation att konsumera enorma resurser. T.ex. en liten bit av järn kan ha mer än 1×1026 atomer! Lyckligtvis är det inte nödvändigt att lösa en såstor Schrodinger-ekvation om vi vill undersöka en egenskap. Täthetsfunktionsteori (DFT) ger oss ett sätt att kosta mindre och fånästan samma. DFT är inte helt korrekt eftersom vissa nödvändiga fören- klingar ingi praktiska beräkningar. DFT kan ge oss goda förutsägelser om egenskaper sstruktur, magnetism och spektrum inom msystem. DFT utveck- lar fortfarande mmetoder och de kommer att göra DFT kraftfullare och mer exakt. I denna tesen studerades den multiferroa BiFeO3 med DFT-metod. Prefixet "ferro"betyder att ett material har en särskild egenskap i avsaknad av det ex- terna tillstsom är nödvändigt för att denna egenskap ska kunna förekomma i andra icke-fero material. Exempelvis innehmagnet en ferromagnetisk egenskap eftersom den visar magnetism i frav yttre magnetfält, medan en aluminiu- minte innehen ferromagnetisk egenskap eftersom den endast visar magnetism om den befinner sig i ett yttre magnetfält. Multiferroic betyder att ett material som har tvåeller flera egenskaper som verkar pådet ferroiska sättet. BiFeO3 har bferromagnetisk och ferroelektrisk egenskap. BiFeO3 har icke-noll elektrisk polarisering i frav ett externt elektriskt fält samt ett icke-nollmagnetiskt moment i frav ett externt magnetfält. Multiferroics är intressant, eftersom samexistensen av tvåferroparametrar inte är vanligt och samspelet mellan de tvåparametrarna kan ge oss nya sätt att använda material, skontroll av mag- netiseringen av datalagringsenheten genom ett tillämpat elektriskt fält. I syn- nerhet har jag studerat strukturparametrar, elektronisk bandstruktur, elektrisk polarisering och magnetisk egenskap hos BiFeO3.

12 2. Density Functional Theory

2.1 Many Body problem The main purpose of condensed matter physics is to describe the behavior of electrons in atoms, molecules and solids. The most fundamental equation describing non-relativistic electrons-nuclei system is the Schrödinger equation

HΨ = EΨ (2.1) Ψ is the total wave function of all ions and electrons in the system and E is the total energy of the system. H is the Hamiltonian which contains kinetic terms and interaction energy terms of both nuclei and electrons, it can be expressed as 2 2 2 2 2 h¯ 2 1 ZIZJe h¯ 2 1 e ZIe H = −∑ ∇I + ∑ −∑ ∇i + ∑ −∑ I 2mI 2 I6=J |RI − RJ| i 2me 2 i6= j |ri − r j| i,I |RI − ri| (2.2) where RI, ri represent the positions of nuclei and electrons, mI and mI represent the mass of the nuclei and electrons respectively. ZI is the atomic number.In this Hamiltonian, the first term is the kinetic energy of all nuclei, the second term is the Coulomb interaction between nuclei, the third term is the kinetic energy of all electrons, the fourth term is the Coulomb interaction between electrons and the last term is the Coulomb interaction between electrons and nuclei. Solving the exact solution of the Hamiltonian of all nuclei and electrons is almost impossible. The most simple but powerful approximation to be made is the Born-Oppenheimer Approximation. The mass of electron is only 1/1836 of that of proton, so they are much lighter than nuclei and move much faster than nuclei. Therefore the electrons respond almost instan- taneously to any movement of nuclei, the nuclei are almost static compared to electrons. This approximation makes it possible to separate the nuclei part from the electronic part in the Hamiltonian. The Hamiltonian of electrons can be expressed as, 2 2 2 h¯ 2 1 e ZIe H = −∑ ∇i + ∑ − ∑ (2.3) i 2me 2 i6= j |ri − r j| i,I |RI − ri| The problem is simplified to work out the electronic wave function ψ(R,r) which is governed by the kinetic energy, electron-electron Coulomb interaction and the static Coulomb potential field of nuclei. However, this Schrödinger equation is still unsolvable due to the complexity of the electron-electron Coulomb interaction. Thus a better method is needed to describe electrons.

13 2.2 Kohn-Sham equation According to Hohenberg-Kohn theorems[1], for an interacting electrons in an external potential Vext Theorem 1 The external potential Vext can be determined by the ground state density n0(r) except for a constant. Theorem 2 The density n(r) that minimizes the total energy functional E[n(r)] is the exact ground state density n0(r). These theorems don’t tell us the exact form of the energy functional E[n(r)]. Kohn and Sham (1965) derived single-particle Schrödinger equation by variational principle. Following their approach, we can write the total energy functional E[n] as Z 1 Z e2n(r)n(r0) E[n] = T[n] + V (r)n(r)dr + drdr0 + E [n] (2.4) ext 2 |r − r0| xc in which T[n] is the kinetic energy functional of the hypothetical non-interacting electrons, Vext is the external potential due to the nuclei or ions, the third term is the Coulomb (Hartree) energy, and Exc[n] is the exchange-correlation energy which includes all many-body effects. The exchange-correlation part of kinetic energy Txc is included in Exc We can use the variational principle on the total energy functional E[n] and the minimization of the energy functional results in the Kohn-Sham(KS) equation,

h¯ 2 [− ∇2 +V (r)]φ (r) = ε φ (r) (2.5) 2m e f f i i i where the effective potential Ve f f is deﬁned as Z e2n(r0) V (r) = V (r) + dr0 +V (r) (2.6) e f f ext |r − r0| xc where the exchange-correlation potential is expressed as

δExc[n] Vxc = (2.7) δn(r) The density is N 2 n(r) = ∑|φi(r)| (2.8) i

If the exact form of Exc is given, the ground state energy E(n0) can be obtained from the Kohn-Sham approach. It should be noted that the eigenvalues εiof the Kohn-Sham orbital φi have no signiﬁcant physical meaning and the sum of these energy eigenvalues does not equal to the total energy but is related as

N Z 2 0 Z 1 e n(r)n(r ) 0 δExc[n] ∑εi = E + 0 drdr − Exc[n] + n(r)dr (2.9) i 2 |r − r | δ[n(r)]

14 2.3 Exchange correlation functionals The DFT theory successfully separates the single particle kinetic energy and the Coulomb (Hartree) energy from the many-body exchange-correlation functional, which can be expressed in many different approaches, such as the local spin density approximation (LSDA) and the generalized gradient approximations (GGA).

2.3.1 Local spin density approximation (LSDA) Hohenberg and Kohn already suggested the local density approximations in their ﬁrst DFT paper[1]. They pointed out that electrons in solids can be of- ten considered as homogeneous electron gas. The LDA exchange-correlation functional has a quite simple form. Z LDA Exc [n(r)] = n(r)εxc(n(r))dr (2.10)

Where εxc(n) represents the exchange-correlation energy density of a homogeneous electron gas with density n(r). The local spin-density approximation (LSDA) is a generalization of LDA formulated in terms of two spin densities n ↑ (r) and n ↓ (r). Z Exc[n↑,n↓] = n(r)εxc(n↑,n↓)dr (2.11)

The exchange-correlation energy Exc can be decomposed into exchange and correlation parts, Exc[n(r)] = Ex[n(r)] + Ec[n(r)] (2.12)

The analytic form of εx term of homogeneous electron gas can be derived from the Dirac’s work in 1930[17].

Z 4 Ex[n(r)] = −k n 3 (r)dr (2.13)

1 1 1 3 3 3 − 3 3 3 3 where k = 2 ( 4π ) for LSDA and k = 2 2 ( 4π ) for LDA. Unfortunately, we only know analytic expressions for the correlation part Ecin the high[18][19] and low[20] density limits. A commonly used from is the interpolation formula of Perdew and Zunger [21] of which interpolation coefﬁcients are derived from the date of quantum Monte Carlo of the homogeneous electron gas generated by Ceperley and Alder[22]. Despite LDA’s simplicity, it gives good predictions for system with slowly varying charge densities. LDA’s prediction of lattice constants is accurate to within a few percent. But LDA has several deﬁciencies, it tends to give higher binding energy. In magnetic system, LDA may gives wrong prediction of magnetic order, for example Fe is predicted to be FCC paramagnetic by LDA, but

15 it has bcc ferromagnetic order in experiment. In strongly correlated systems where particles interact with each other, LDA gives inaccurate result. For instance, LDA predicts transition metal oxides FeO,CoO, NiO and MnO to be a metals or semiconductors, but they are all Mott insulators. With the help of Hubbard-corrected functionals, the LDA+U method which will be discussed in latter chapter, the insulating state of FeO was obtained[23].

2.3.2 Generalized gradient approximation (GGAs) LDA has bad performance is system with rapidly changing charge density. An improvement can be easily considered is to include the gradient of the electron density, then we have the generalized gradient approximations (GGA). The general form of GGA is: Z GGA Exc [n(r)] = f (n(r),∇n(r))dr (2.14)

Most GGAs are based on corrections on LDA, the gradient of electron density ∇n(r) can be considered as the effect of the velocity of electrons’ movement. The PBE[24] is a commonly used form of GGA, in which all parameters are constants, it is a simpliﬁcation of the PW91[25]. Comparing with LDA, GGAs correct the overestimated binding energy , give correct prediction for magnetic systems such as Fe’s BCC ferromagnetic order and perform better for bulk phase stability ect.. However, similar to LDA, GGAs are inaccurate to describe band gap of transition metal and rare earth compounds.

2.3.3 Beyond GGA In meta-GGA, the second order gradient, Laplacian, is included in the functional. The Laplacian ∇2n can be considered as the effect of the kinetic energy of electrons’ movement. It has be shown that accurate band gaps could be obtained for many materials with meta-GGA[26]. Some functionals always overestimate energy some others always underestimate energy, people may ask what if we use the combination of functionals. That is the idea of hybrid functionals. Most hybrid functionals take part of Hartree-Fock energy, LDA energy and GGA energy into the exchange- correlation energy, for example the most successful functional for chemistry B3LYP [27]: B3LYP LDA HF B88 Ex = 0.8Ex + 0.2Ex + 0.72∇Ex (2.15)

B3LYP VMN3 LYP Ec = 0.19Ec + 0.81Ec (2.16) B88 Where Ex is the exchange part of the Becke 88 exchange functional[28], the VMN3 Ec is the correlation energy of Vosko-Wilk-Nusair correlation functional

16 LYP III[29] and Ec is part of Lee, Yang and Parr correlation energy [30]. The famous Jacod’s Ladder

2.3.4 L(S)DA+U: On-site Coulomb-repulsion "The quaint acronym "LDA+U[31]" stands for methods that involve LDA- or GGA-type calculations coupled with an additional" orbital-dependent interaction. The strong on-site Coulomb interaction of localized electrons is not correctly described by LDA or GGA, which makes the L(S)DA and GGA un- able to describe localized d and f orbitals. This deﬁciency is remedied by introducing the Coulomb interaction U into the LDA energy functional. In our calculations, a simpliﬁed form of LDA+U introduced by Dudarev[32] et al is used. In Dudarev’s form, the energy functional is of the form:

U − J σ σ σ ELDA+U = ELDA + ∑[(∑nˆm1,m1) − ( ∑ )nˆm1,m2nˆm2,m1] (2.17) 2 m1 m1,m2 In which n is the density matrix element, σ represents the spin state. In Du- darev’s approach only the effective U value Ue f f = U − J is of signiﬁcance.

2.4 Plane wave sets and pseudopotential In periodic structure, the wave function is in the form of Bloch wave: ψ(r) = eikru(r) (2.18) where u(r) is a periodic function with same periodicity as the crystal. Any periodic function can be expanded with a discrete set of plan wave functions by Fourier transformation method. Thus all wave functions can be expanded with a discrete plan-wave basis set in periodic lattice. Only a basis set with infinite functions can expand a wave function completely. However, a finite basis set with a suitable energy cutoff is enough to expand a wave function precisely. That is because functions with lower energy (lower frequency) are of more importance than functions with higher energy (higher frequency). The absence of high frequency part will cause error in total energy, thus in calculation the energy cutoff should be high enough to make the error smaller than the tolerance. Since the wave function in core region oscillate rapidly, a plane-wave basis set with finite cutoff energy usually fails in expanding core orbitals and wave function of valence electrons in the core region. In the pseudopotential approximation[33], a real atomic potential is replaced by a more smooth pseudopotential which make it easier to be expanded by plan-wave basis. The valence electrons are more essential to physical properties of solid crystals than core electrons, since chemical bonding happens in the inter-core region. The pseudopotential is smoother than atomic potential in the core region

17 Figure 2.1. Schematic representation of pseudo-potential and pseudo-wavefunction, the pseudo-potential and pseudo-wavefunction are in dashed lines. The core region is deﬁned as r ≤ rc . and is identical to atomic potential out of the core region, which makes the pseudo wave function nodeless in core region and easier to be expanded by plan-wave basis. A schematic representation of pseudopotential is shown in Fig. 2.4. The pseudopotential introduced by Hamman, Schluter and Chiang was constructed from norm conserving pseudo potentials (NCPP). The nodeless pseudo- wavefunction ψpseduo conserves norm and energy eigenvalues of the true all electrons wave function. The NCPP gives accurate results but still requires much effort to calculate. The projector augmented wave (PAW) method was introduced by Blochl¨ [34]. It’s a general combination of the linear augmented wave method and the pseudopotential method. The main goal of this method is to avoid using the oscillatory true atomic wave functions when calculating physical properties like expectation values and densities. The APW method starts from expanding an true oscillatory all-electrons single particle KS wave function ψn with a set of smooth pseudo wave functions ψ˜n by a linear transformation T:

|ψni = T |ψ˜ni (2.19)

In the following, a means the function or operator is in the augmentation sphere around the a atom. The transformation operator T should be in this

18 form: T = 1 + ∑Ta (2.20) a in order to make the all-electrons wave function and smooth pseudo wave function identical out the core region. In another word, T only transform wave function in the core region. Each core region is defined in a augmentation a sphere with cut-off radius rc around an atom as: a a |r − R | ≤ rc (2.21) The solutions to the Kohn-Sham Schrodinger equation for an isolated atom are partial waves which also form a complete basis. T transforms the partial waves φi to the smooth partials waves φ˜i. a ã |φi i = T φi (2.22) In augmentation spheres the smooth pseudo wave function is expanded into smooth partial waves: a ã |ψñi = ∑Cni φi (2.23) i Applying the transformation T to the left of the both sides of equation (2.23), we found that the all-electron wave function ψn is expanded into partial waves a a φi by the same parameters Cni. a ã T |ψñi = T ∑Cni φi (2.24) i

a ã T |ψñi = ∑CniT φi (2.25) i a a |ψni = ∑Cni |φi i (2.26) i a The expanding coefficients Cni can be expressed as the projection of smooth wave function ψn onto projector functions pi. a ã Cni = pi ψñx (2.27) ã The projector functions pi are of completeness and orthogonality in the augmentation spheres. ã ã ∑ φi pi = 1 (2.28) i ã ã pi φi = δi, j (2.29) By using the completeness and orthogonality, we can derive the expression of T: a ã a T = 1 + ∑∑(|φi i − φi )hpi | (2.30) a i

19 a The transformation T is deﬁned by all-electrons partial waves φi , smooth ˜a a pseudo partial waves φi and projector functions pi . The expectation value of any operator Oˆ is calculated by the inner product:

oi = hψ|Oˆ |ψi (2.31)

The expectation oi can be expressed as the inner product of smooth pseudo wave function and operator. † oi = hψ˜ |T OTˆ |ψ˜ i (2.32) where T †OTˆ can be deﬁned as a smooth pseudo operator Oˆ˜. The frozen core approximation is always used with the PAW method. In this approximation, the core states of the isolated atoms are assumed not affected by the formation of chemical bonds. Only valence electrons orbitals are considered in this approximation, which reduce the computation cost signiﬁcantly.

2.5 First principles phonon calculations The harmonic approximation is commonly used in phonon frequency calculation. Atoms are assumed oscillating around their equilibrium positions r with displacements d. The potential energy function Φ is assumed to be a function of the displacements around equilibrium positions up to the second order.The potential is expressed in the following series form:

1 0 0 0 0 Φ = Φ0 + ∑ ∑Φa(ln)da(ln) + ∑ ∑ Φab(ln,l n )da(ln)db(l n ) a=x,y,z ln 2! a,b ln,ln0 (2.33) where a, b=x, y, z represent directions of three Cartesian coordinate axes and l and n are the labels of unit cell and atom in the unit cell respectively.Φ0 is the zeroth order force constants which does not depend on the positions of atoms, th it is set to be 0 usually. Φa(ln) is the first order force constant acting on the n atom in the lth unit cell along a direction which only depends on the position of 0 0 one atom and one direction.Φa,b(ln,l n ) is the more complicate second order force constant which is determined by two atoms and two directions. The finite displacement method is used to calculate these force constants. Potential energies of unit cells with small displacements along different directions at constant volume are calculated by first principle method such as DFT 0 0 method. A force Fa(ln)and a second-order constant Φa,b(ln,l n ) are obtained by partial derivatives of potential energy. −∂Φ Fa(ln) = − (2.34) ∂da(ln)

2 0 0 0 0 ∂ Φ ∂Fb(l n ) Φa,b(ln,l n ) = 0 0 = − (2.35) ∂da(ln)db(l n ) ∂da(ln)

20 With the ﬁnite displacement method,the force is given approximately by the potential energy difference between the unchanged and displaced unit cell as

V −V da(ln) Fa(ln) = (2.36) da(ln) and the second-order derivative is also replaced by

0 0 da(ln) 0 0 0 0 Fb(l n ) − Fb (l n ) Φa,b(ln,l n ) = (2.37) da(ln)

th where the superscript da(ln) means the n atom is displaced d along the a direction. The force on atoms at equilibrium positions are all zero. With these second-order force constants the dynamical matrix D(q) is constructed as

0 0 Φa,b(0n,l n ) ik[r(l0n0)−r(0n)] D 0 (k) = √ e (2.38) ab,nn ∑ 2 l0 mnmn0 where the .The dynamical matrix describes interaction between the nth atom 0th with a mass of mnin one unit cell and the n atom with a mass of mn0 in all unit cells. The sum of the unit cells usually runs over the nearest neighbors in practical calculation in order to reduce the cost. The phonon frequency ωk and polarization vector pk are obtained by solving eigenvalue equation of dynamical matrix D(k),

2 ∑Dab,nn0 (k)pk j,bn0 = ωk jpk j,an (2.39) bn0 where j is the index of phonon band. The displacement vector of the nth atom in the lth unit cell can be derived from its corresponding polarization vector,

A iqr(ln) d(ln) =√ pk j,ne (2.40) mn where A is a complex constant. The displacement vectors are used for analyzing and visualizing the vibra- tion modes. The phonon dispersion is plotted from phonon frequencies at k points connecting high-symmetry points in reciprocal space. The potential energy Φ is at its minimum if the crystal is in ground state, which means any displacement of atom from equilibrium position increase the energy. The phonon frequencies of a stable phase are real and positive at all k points. The imaginary frequencies (always shown as negative frequencies in dispersion curve) means the current unit cell is not the ground state and has a trend of phase transition.

21 3. Modern polarization theory

3.1 Polarization lattice Electric dipole moment is a quantity measuring how far a positive point charge and a negative point charge are separated from each other. The electric dipole moment has a magnitude p = qd, where q is the charge and d is the distance between two charges. The q points from the negative charge to the positive charge. For a collection of charges,it is deﬁned as

p = ∑qnrn (3.1) n where qn is the charge and ri is a vector from reference point to the charge qn. For the case of continuous distribution, the total electric moment is defined as: Z p = q(r)rd3r (3.2) where r is the position vector relative to the reference point of the system and q(r) is charge density at r. In microscopic and finite system such as molecule and cluster, the above definition 3.2 works perfectly, different reference points will always give the same result. As shown in Fig.3.1, a chain with the positive and negative charges spaced a distance a/2 apart and so the length of unit cell is a. This chain has inversion symmetry to point charge in it so this chain system is non-polar and with a dipole moment density of zero. In macroscopic scale, the dipole moment density is used to describe the polarization. For crystal materials, the dipole moment density can be defined as the dipole moment per unit cell. However, the choice of unit becomes a trouble when we are try to use the formula 3.2 to calculate the dipole moment density even for the most simple case, one-dimension chain of alternative positive charges and negative charges. Two choices of unit cell are possible for

Figure 3.1. A chain of evenly placed anions and cations

22 this system, in one the negative charge sits left to the positive one, in another one the negative one sits right to the positive one. Now,let us calculate the dipole moment per unit cell, the reference point is chosen as the center of unit cell. The dipole moment density for one choice of unit cell is

p = (−q × (−a/4) + q × a/4)/a (3.3) = (qa/4 + qa/4)/a (3.4) = q/2 (3.5)

The dipole moment density for another one choice is

p = (q × (−a/4) + (−q) × a/4)/a (3.6) = (−qa/4 − qa/4)/a (3.7) = −q/2 (3.8)

We get two different value for the identical system, and neither of them is zero. We will get even more values if we make a dipole consist of a positive charge and a farther negative charge but not the neighbor one. For this chain these values are q(1 + n)/2, n is an arbitrary integer. This set of dipole moment density values are called polarization lattice. For non-polar system, the values in the lattice should be symmetric to the origin point but the origin itself is not necessary to be in the lattice. In solid state crystal structure , if we let anions combined with electrons of other anions as dipoles we will also get a polarization lattice. The lattice constant of polarization lattice is called polarization quantum. Then let us calculated polarization again for a polar chain in which all positive charges moves distance d relative to the negative charges as shown in Fig.3.1. In this distorted chain, we will ﬁnd out our left part is different with the right part if we sit on a charge, so this system must be polar and should have an non-zero value of dipole moment density. the dipole moment density for one choice of unit cell is

p = [−q × (−a/4) + q × (a/4 − d)]/a (3.9) = (qa/4 + qa/4 − qd)/a (3.10) = q/2 − qd/a (3.11)

The dipole moment density for another one choice is

p = [q × (−a/4 − d) + (−q) × a/4]/a (3.12) = (−qa/4 − qd − qa/4 + qd)/a (3.13) = −q/2 − qd/a (3.14)

We can extrapolate that the values of polarization lattice are q(1+n)/2−qd/a, so the values of lattice are not symmetric to the origin which is a signature of

23 Figure 3.2. A distorted chain of anions and captions polar system. What is the polarization of this distorted chain? We will answer −qd/a without thinking. In conclusion, the absolute values of polarization has no physical meaning, the change in polarization between non-polar and polar system is the value we can measure in experiments.

3.2 Berry phase theory for the macroscopic polarization

The electronic contribution to the difference in polarization Pel can be treated as a berry phase of the valence wave function[35][36]. The change in polarization along a transformation pass from non-polar structure to polar structure is polar non−polar ∆Pe = Pe − Pe (3.15) with N Z i f |e| 3 Pe = − ∑ hunk|∇k |unkid k (3.16) Ω0 n=1 BZ where Ω0 is the volume of unit cell, f is the occupation number of valence states, N is the number of occupied bands, unk is the cell-periodic part of the Bloch function ψnk. It’s more understandable in the form of localized Wannier functions of occupied bands f |e| N Pe = − ∑ hWn|r|Wni (3.17) Ω0 n=1

Where Wn is the Wannier function associated with band n. The Wannier center is deﬁned as rn = hWn|r|Wni (3.18) The Wannier center is the average position of the electrons in the Wannier function. It is noteworthy that polarization values obtained from calculations will sit on several branches spaced by polarization quantum, only the change of polarization on the same branch is the true value of polarization.

24 4. Magnetism

4.1 Heisenberg Hamiltonian and Magnetic ordering The Heisenberg Hamiltonian for two electrons system is written as

H = −2J12S~1 · S~2 (4.1) where J12 is the exchange constant between the S~1 and S~2, J12 = J21 . If J12 > 0, the triplet state

|1,1i = ↑↑ (4.2) √ |1,0i = (↑↓ + ↓↑)/ 2 (4.3) |1,−1i = ↓↓ (4.4) with a symmetric spin function and an antisymmetric spatial function has lower energy than the single state √ |0,0i = (↑↓ − ↓↑)/ 2 (4.5) with a antisymmetric spin part and an symmetric spatial part. So the two electrons system has a total spin moment S=1. If J12 < 0, the single state is ground state. So the system has a total spin moment S = 0. In solids, the Heisenberg Hamiltonian is expressed as the sum of exchange Hamiltonians of all pairs of atoms:

1 ~ ~ ~ ~ H = (−2∑Ji jSi · S j) = −∑Ji jSi · S j (4.6) 2 i, j i, j

The 1/2 factor is introduced because the exchange interaction between any pair is counted twice in the sum. Without external magnetic ﬁeld, if Ji j is positive, parallel arrangement of atomic spins has lower energy and the magnetic ordering is ferromagnetic. Without external magnetic ﬁeld, if Ji j < 0, antiparallel arrangement of spins has lower energy and the magnetic ordering is antiferromagnetic. It is convenient to describe antiferromagnet by the concept of sublattice. An antiferromagnetic lattice can be divided into one sublattice with all spin up atoms and one sublattice with all spin down atoms. The magnetic moments of the two sublattice in antiferromagnet have opposite directions and same magnitude. In ferrimagnet, the two magnetic moments of the two sublattices still hold opposite directions but have different magnitudes.

25 Figure 4.1. Schematic representation of two-dimension magnetic order

An example of two-dimension ferromagnet, antiferromagnet and ferrimagnet is shown in fig.4.1 The total magnetic moments of ferromagnet and ferrimagnet are non-zero, the total magnetic moment of antiferromagnet is zero. The spontaneous magnetic orderings discussed above will be broken by thermal disturbance, the Currier temperature TC is the temperature above which cer- tain materials lose their permanent magnetic ordering. All magnetic orderings will transform to paramagnetic phase above TC. The total magnetic moment of paramagnet depends on the applied external magnetic field as shown in Fig.4.1 The paramagnet shows zero magnetic moment in the absence of applied magnetic field. In the presence of applied magnetic field, paramagnet has the same magnetic moment direction as the external magnetic field.

4.2 Magnetic anisotropy A system conserving its total energy under the change of magnetization direction is magnetic isotropic, otherwise it is magnetic anisotropic. The magnetic anisotropy energy(MAE) is deﬁned as the energy difference between the ground state and energetically unfavorable conﬁguration commonly and it’s used to evaluate how the intensity of anisotropy. The magnetic anisotropy is a result of the breaking of the rotational symmetry, which can be caused by dipole-dipole interaction and spin-orbital coupling. The spin-orbital coupling yields the magnetocrystalline anisotropy, in which we are interested.

26 4.2.1 Single-ion anisotropy and spin-orbital coupling In quantum ﬁeld theory, the spin is given naturally from the Lorentz invari- ance. The wave equation of relativistic electron is the Dirac equation. In v the non-relativistic approximation ( c 1), the Dirac equation is reduced to a Schrödinger equation with a revised Hamiltonian,

p2 eS · B p4 eh¯ 2 e H = − eΦ + − − ∇2Φ − S · (∇Φ×p) (4.7) 2m m 8c2m3 8c2m2 2m2c2 where the last term is spin dependent. The potential function Φ is approximately spherically symmetric, so the its gradient is pointed to the radical direction. r dΦ ∇Φ = (4.8) r dr The the spin dependent term Hsoc takes the following form, e dΦ H = − S · (r×p) = ξL · S (4.9) soc 2c2m2r dr where ξ is the spin-orbital coupling constant which depends on the type of atoms. The spin angular moment is coupled to the crystal lattice via the orbital angular moment. The spin-orbital coupling can be also described in a classical picture. For an particle in a potential field, the spin-orbital coupling is the relativistic interaction between the magnetic moment µ associated with its spin and the magnetic field H generated by the orbital motion of itself in the potential. The magnetic field always tends to align the spins to its own direction, then the free energy of the crystal depends on the angle between he magnetic moment µ and the magnetic field H.

4.2.2 Non-collinear magnetism and Dzyaloshinskii-Moriya interaction The spin-orbital coupling is the origin of the single-ion anisotropy for collinear spins. In the non-collinear spins conﬁguration, part of the magnetic anisotropy is caused by the anisotropic superexchange originating from the Dzyaloshinskii- Moriya (DM) [37][38] interaction. The Hamiltonian of the DM interaction takes the following form,

HDM = ∑ Di j · (S1 × S2) (4.10) i6= j where Di j is the DM vector. Unlike the Heisenberg exchange interaction the DM interaction is antisymmetric, the swap of the two spins change the sign of the vector product. The DM interaction decreases the total energy when

27 the angle between DM vector and the product of two spins is obtuse. The DM interaction makes the spins favor canted conﬁguration, which can cause weak spontaneous magnetization in anti-ferromagnetic material[39] and spin- spirals[40]. The microscopic mechanism of DM interaction is identiﬁed as spin-orbital coupling by Moriya[41]. The DM interaction between two neighbor spins is transferred by a third atom which is called superexchange mechanism.

28 5. Results and discussions

5.1 Computational details The first-principles calculations were performed in the framework of spin- polarized density functional theory (DFT) using the projector augmented wave (PAW)[42][43] method and a plane-wave basis set as implemented in the Vi- enna ab initio simulation package (VASP)[44][45].The cutoff energy for the plane waves was set to be 500eV. Perdew-Bruke-Enzerhof’s (PBE) version of the generalized gradient approximation (GGA)[24] was used to describe the exchange correlation density functional. For better descriptions of the electronic and magnetic properties, 15 valence electrons were treated for Bi (5d106s26p3), 14 valence electrons for Fe (3p63d64s2), 6 for O (2s22p4). Pro- jection operators were evaluated in reciprocal-space. The energy tolerance of electronic steps calculations was 10−7 eV. It’s reasonable to set the G-type (Rock-Salt) antiferromagnetic (AFM) order as the initial magnetic configuration in all calculations. The experiments showed BiFeO3 has a approximate G-type AFM order modified with long- wavelength spin spiral structure[8] and total residual magnetic moments caused by weak ferromagnetism[46][12]. In geometry relaxations,a 5×5×5 Monkhorst-Pack k-grid mesh [47] centered at Γ was used to sample the Brillouin zone. The Fermi smearing method was employed for the total energy calculations with the width of 0.05 eV. The atomic positions were fully relaxed using the conjugated gradient method for the energy minimization with a criterion that requires the force on each atom smaller than 0.001 eV/Å. A well known defect of DFT calculation is that it always underestimate the band gap of system with localized d orbitals, which can be remedied by introducing on-site Coulomb repulsion between d states in DFT+U[31] approach. In this work, the GGA+U method was implemented in The simplified approach was introduced by Dudarev et al[32].In this approach only the effective Hubbard parameter Ue f f = U −J is meaningful, thus the J parameter was set to be zero in all calculations. The Ue f f for Fe 3d states was set to be 0 eV, 2 eV and 4 eV. The electronic polarization was calculated by the Berry phase theory[35][36]. These calculations were performed with a 8×8×8 k-point mesh,the number of k-points on the string was set to be 8. The center of the cell in direct lattice coordinates which the total dipole-moment of the cell is calculated is specified in a way that all ions keep in the same side. The ionic contribution to polarization is calculated by summing the product of the position of each ion with the

29 charge of each ion’s core. The total polarization is the sum of electronic polarization and the ionic polarization. The R3C¯ phase (centrosymmetric phase) of BiFeO3 is a candidate for paraelectric phase in high temperature. We used the R3C¯ phase with the same cell volume as R3C phase as the midpoint between R3C phase and its enantiomorphic counterpart -R3C phase. The transition path is generated by linear interpolation from R3C phase to R3C¯ phase in our calculations. To investigate the non-collinear magnetism of BiFeO3, magnetic anisotropy energy of BiFeO3 was calculated in two approaches: non-self-consistent and self-consistent approach. The spin-orbital coupling was switched on in all calculation. In the self-consistent approach, the residual total magnetic moment and canted alignment of magnetic moments were obtained. The MAE calculations were done with k-points mesh in different sizes to study the convergence of MAE. The electronic step convergence threshold of 10−7eV was accurate enough for the small value of MAE. The spin-spiral dispersion calculations were performed in the ﬁrst Brillouin zone along [111] and [10-1] direction of the rhombohedral lattice.

5.2 Structure of BiFeO3

The ground state of BiFeO3 in room temperature has a rhombohedral symmetry and R3c space group. The R3c primitive unit cell is derived from the cubic perovskite by rotating the two neighbouring FeO6 octahedra about the [111] axis in the opposite ways and displacing ions along [111] axis. The ideal perovskite lattice has a Pm3¯m space group. The rotation of FeO6 alone reduces the symmetry from Pm3¯m to centrosymmetric R3¯c space group. While the displacement along [111] reduces the symmetry from Pm3¯m to rhombohedral R3m. The combination of these two deformations lowers the symmetry to R3C and makes BiFeO3 polarized and highly distorted. In Table.5.1 the important structural parameters of BiFeO3 R3c phase obtained from geometry relaxation are shown for Ue f f = 0, 2, 4 eV. Comparing with calculated parameters from Neaton et al’s [12] work and Ravindran et al’s[16] work and measured data from Kubel et al’s[48] work, the effective U value of 2 eV is found to be most appropriate for geometry optimization. The volume of primitive cell is under- estimated by the LSDA in the work of Neaton et al [12] and is overestimated by GGA in this work and the work of Ravindran et al[16].The value of U affects parameters of BiFeO3 to a very limited extent. With U varying from 0eV to 2eV, the lattice constant a is increased only by 0.24%. Noticeably the structural parameters of R3c BiFeO3 do not increase monotonically with U value, which means there is no linear relation between the structure and the on-site Coulomb repulsion. In the geometry relaxation of R3¯c phase, the cell and atom position are relaxed together.

30 ◦ 3 Structure a(Å) α( ) V0(Å ) Bi(2a) Fe(2a) O(6b)

R3c, Ue f f =0 eV 5.684 59.229 127.60 0 0.223 0.564, 0.966, 0.387

R3c, Ue f f =2 eV 5.698 59.048 127.99 0 0.221 0.566, 0.966, 0.386

R3c, Ue f f = 4 eV 5.697 58.989 127.74 0 0.219 0.568, 0.966, 0.386 R3c (Exp, Ref.[48]) 5.63 59.35 124.60 0 0.221 0.539, 0.933, 0.395

R3c (theory, Ref.[16]) 5.697 59.235 128.48 0 0.2232 0.534, 0.936, 0.387

R3c (theory, Ref.[12]) 5.50 59.99 117.86 0 0.228 0.542, 0.943, 0.397

R3c,¯ Ue f f =2 eV 5.511 61.347 121.92 R3c¯ (theory, Ref.[12]) 5.513 61.432 122.29

Table 5.1. The structural parameters of BiFeO3 from calculations and experiments. The Wyckoff position parameters represent positions of Bi (x, x, x), Fe (x, x, x) and O (x, y, z) atoms in rhombohedral system. The lattice constant of the rhombohedral primitive cell a is given in Å. The α is the rhombohedral angle, a rhombohedral primitive cell with α=60◦ is equivalent to an ideal cubic perovskite.

In Table5.1, the structure of R3¯c phase from this study is very close to the result from the work of Neato et al[12]. The rhombohedral angle α of R3c ◦ ◦ phase was 59.048 from the calculation with Ue f f = 2eV is smaller than 60 , which indicates its primitive cell is elongated along the [111] axis comparing with the ideal perovskite. Whereas the primitive cell of R3¯c is compressed along the [111] axis with α larger than 60◦. The schematic representation of primitive cell of R3c phase is shown in Fig5.1. The two FeO6 octahedra were rotated along [111] in opposite directions which makes the [111] a C3 rotation axis. From the left panel of Fig.5.1 we can see that the center of the O6 does not overlap with the Fe atom sur- rounded by O atom. The center of the O6 is 0.27 Åcloser to the Bi(0.5, 0.5, 0.5) than that Fe atom. The O6 is no longer an ideal octahedron in R3c BiFeO3, the distance between ﬁrst neighbouring O atoms varies from 2.76 to 3.04 Å. The primitive cell of R3¯c BiFeO3 is less distorted than R3c phase, the center of O6 is overlapped with the Fe centered in the octahedron as shown in Fig.5.2. The O6 is still not an ideal octahedron, the distance between neighbouring O atoms has two values: 2.93 Å and 2.79 Å even without ions displacement. This small distortion on O6 octahedron possibly originates from the chemical environment’s asymmetry caused by the rotation of FeO6 about [111].

31 Figure 5.1. Schematic representation of the structure of R3c BiFeO3 from GGA+U (Ue f f =2eV) calculation viewed from the standard orientation of crystal shade (left side) and [111] axis direction (right side).

Figure 5.2. Schematic representation of the structure of R3¯c BiFeO3 from GGA+U (Ue f f = 2eV) calculation.

32 Figure 5.3. Electronic band structure of R3c BiFeO3 for Ue f f =0, 2 ,4 eV.

5.3 Electronic properties of BiFeO3

The electronic band structure of ferroelectric R3c BiFeO3 is presented in the Fig.5.3. The R3c BiFeO3 is insulating with a gap of 1.03 eV in the GGA calculation. The band gap is increased to 1.83 eV the Ue f f value of 2 eV and increased to 2.22 eV by Ue f f =4 eV. The R3c BiFeO3 has an direct band gap in GGA calculation at the Z point. The band gap becomes indirect in GGA+U calculations, the bottom of conducting band locates between the F point and Γ point, the top of valence band locates on the Z point. The total density of states (TDOS) from GGA and GGA+U calculations for R3c BiFeO3 are shown in Fig.5.4. The spin-up channel and spin-down channel are symmetric to each other for TDOS of BiFeO3 and Fe atoms which is a sign of the antiferromagnetic order of the R3c BiFeO3. We can see that the band gap is increased by implementing GGA+U calculations in the Fig.5.4. The projected density of states (PDOS) of Bi, Fe and O atoms of R3c phase of BiFeO3 in GGA+U calculation with Ue f f = 2 eV is shown in the left panel of Fig.5.5. The valence bands of R3c in the range from -6 eV to Fermi level are mainly composed of the 2p orbitals of O atoms and 3d orbitals of Fe atoms. The hybridization of Fe 3d band and O 2p band is related to the formation of FeO6 octahedron. The band with lower energy locating in the range from -10

33 Figure 5.4. Total density of states of BiFeO3 and Fe, Bi and O atoms in R3c BiFeO3 for Ue f f = 0, 2, 4 eV.

eV to -8 eV is the 6s band of Bi atoms. Bi 6s band has no hybridization which is consistent with the existence of Bi lone pair[49] in BiFeO3. In the atomic orbital of Bi, the 6p orbitals has higher energy than the 6s orbitals. But the 6p bands in R3c BiFeO3 almost disappear in the valence band below Fermi level. Obviously, the reason is that each Bi atom lose 3 6p electrons to O atoms and then becomes Bi3+ ion with only 6s electrons in its valence band in the R3c BiFeO3. Further, the acquiring of electrons can account the abundance of 2p states of O atoms in the valence band. The magnetic moments of Fe atoms from GGA and GGA+U calculations are in good agreement with the experimental value of 3.75 µB measured from low-temperature neutron-diffraction measurements[8]. The local magnetic moments of Fe atom is ± 3.728, ± 3.980 and ± 4.138 µB respectively for Ue f f = 0, 2 and 4 eV. Since the hybridization between Fe 3d band and O 2p band, the non-magnetic O atoms become magnetic with average magnetic moments per atom of ± 0.072, ± 0.058 and ± 0.045 µB respectively for Ue f f = 0, 2 and 4 e. The O6 octahedron’s magnetic moment has the same direction with the Fe atom it hybridizes with. From magnetic moments from calculations with different Ue f f values, it is found that the introducing of on-sine Coulomb repulsion weakens the hybridization between Fe and O and then increase the magnetic moment of Fe and decrease the magnetic moment of O. The PDOS of R3¯c BiFe3 is shown in the right panel of Fig.5.5. Similar to the R3c phase, the top area of valence band of R3¯c is also mainly composed of the 2p band of O atoms and the 3d band of Fe atoms, and the 6s band of Bi atoms does not hybridize with other band. But, the magnetic conﬁguration of R3¯c phase is ferromagnetic which is totally different with the R3c phase, the Fe atoms have magnetic moments of 1.52 µB . The ferromagnetism of R3¯c causes the asymmetry of the valence band in the PDOS of Fe atoms as

34 Figure 5.5. The projected density of states of Bi, Fe and O atoms of R3c and R3¯c BiFeO3. shown in Fig.5.5. The hybridization between O 2p band and Fe 3d band makes O atoms’ magnetic moments in the same direction as Fe atoms and causes asymmetry of the valence band in the PDOS of O atoms. The 6s band of Bi atoms keeps symmetric which conﬁrms again there is no hybridization exists between Bi 6s band and other bands. Notably, the R3¯c BiFeO3 is not insulting in our calculation. Considering the enlargement effect of U parameters on the band gap of R3c BiFeO3, the conductivity of R3¯c phase may be caused by the most common defect of DFT calculation: it always underestimate the band gap. Thus, the calculation with Ue f f = 4 eV is done to investigate this problem. The TDOS of the R3¯c with Ue f f = 4 eV is shown in Fig.5.6. With Ue f f = 4 eV, the R3¯c BiFeO3 is insulting with a narrow band gap of 1 eV and the magnetic moments of Fe atoms increase to 4.27 µB. Thus, the U value of 4 eV is more appropriate to describe the electronic properties of the paraelectric R3¯c BiFeO3.

5.4 Electric polarization The Berry phase theory[36][50][51] is used to calculate the polarization of R3c BiFeO3. Due to the polarization lattice mentioned in Chapter 3.1, the polarization of R3c BiFeO3 may have several possible values spaced by a ﬁxed polarization quantum. Since the R3¯c is centrosymmetric and paraelectric, it should has a zero polarization in the absence of external electric ﬁeld. Thus, the real polarization value of R3c phase is the polarization difference between the R3c phase and the R3¯c phase.

35 Figure 5.6. Total density of states for the R3¯c BiFeO3 with Ue f f = 4 eV.

The lateral view of the primitive cell of BiFeO3 is shown in the Fig. 5.8. All ions in the primitive of R3¯c are symmetric about the plane perpendicular to [111], whereas the symmetry is broken in the −R3c and R3c. In our calculation, the transition path from R3¯c to R3c was generated by linear interpolation and 3 intermediate structures between R3¯c and R3c were chosen to performed the berry phase calculation. To keep the consistency in our calculation, the primitive cell of R3¯c has the same cell size and shape as the R3c in our berry phase calculation. Not like the R3¯c in Section 5.3 , this R3¯c is insulting with a narrow gap of 1 eV as shown in Fig. 5.7, which is necessary to perform berry phase calculation. This R3¯c BiFeO3 has the AFM order which indicates that the origin of AFM order is not the ions displacement along [111]. The results are shown in Fig. 5.9. The polarization of the R3c BiFe3 is 103.5 µC/cm2 and the polarization quantum is about 123 µC/cm2 calculation. Our result of polarization is consistent with experimental polarization value of 100 µC/cm2 [15] for [111] direction. For the polarization quantum, we see a discrepancy between result from this study and calculation by Neaton et al [12]. One possible reason for the discrepancy is that the centrosymmetric structure in this study is the R3¯C phase whereas the cubic phase is used in others’ calculation. The choice of different transition path is another possible reason. Our calculation conﬁrms the ferroelectric order in R3c BiFe3 and predicts a polarization value close to measured value. To understand the ferroelectricity better and calculate the polarization more accurate, a more reasonable switch- ing path is very necessary.

36 Figure 5.7. Total density of states of the R3¯c BiFeO3 with the same cell size and shape as the R3c BiFeO3

Figure 5.8. The lateral view of the primitive cell of the -R3c, R3¯c and R3c BiFeO3 along the [111] direction

Figure 5.9. Polarization along a path from the centrosymmetric R3¯c structure to the ferroelectric R3c structure calculated with GGA+U method with Ue f f = 2 eV.

37 Figure 5.10. Convergence of MAE and magnetic moments with k points mesh size. MAE = E111 - E1−10

5.5 Magnetic anisotropy The non-collinear magnetic property is reflected in the magnetic anisotropy energy. The spin spiral order is suppressed in our calculation, which is consistent with the case in epitaxial thin films. Firstly, the convergence of MAE in collinear spins and magnetic moments in non-collonear spins with different k points mesh is studied for the antiferromagnetic R3c BiFeO3 from GGA+U (Ue f f =2 eV) calculation. The MAE is defined as the energy difference between two configurations: magnetic moments parallel to [111] and magnetic moments parallel to an axis perpendicular to [111]. As shown in Fig. 5.10, a 5×5×5 k points mesh with 125 points is enough for both MAE and magnetic moments to converge. The single-ion anisotropy is studied by calculate the dependence of energy of the collinear spins on the angle relative to the [111] axis. As shown in Fig. 5.11(a), The (111) plane is found to be the magnetic easy plane, the two spins are opposite to each other and tend to be perpendicular to [111] axis as shown in Fig.5.12(a) . In the (111) plane, the energy does not change with the orientation of collinear spins. The MAE and total magnetic moment of noncollinear spin configuration were calculated self-consistently. As shown in Fig.5.11(b), the noncollinear spins have the same magnetic easy plane the (111) plane as the collinear case, the total magnetic moment also has the maximum in the easy plane. The MAE of non-collinear spins originate from both single- ion anisotropy and anisotropic superexchange, thus the MAE of non-collinear spins is larger than collinear spins. The single-ion anisotropy’s contribution to MAE is 124 µV/unit cell and the contribution of superexchange is 89µV/unit cell. In the non-collinear configuration, M1 and M2 are canted away from the collinear direction as shown in Fig. 5.12, the total residual magnetic moment is 47 mµB/unit cell and lies in the magnetic easy (111) planes. This residual total magnetic moment shows the weak ferromagnetism in BiFeO3. The canting of magnetic moments is caused by DM[37][38] interaction, which is the combination of exchange interaction and spin-orbital coupling.

38 For the system of two magnetic moments this Hamiltonian is written as

HDM = −D · (M1 × M2) (5.1)

The difference of the two magnetic moments is deﬁned as antiferromagnetic vector L = M1 − M2, and the sum of two magnetic moments is deﬁned as the ferromagnetic vector M = M1 + M2. Then the DM Hamiltonian can be written as

HDM = −2D · (L × M) (5.2)

The HDM will reduce the total energy with particular D and then make the two magnetic sublattices canted from collinear direction.

(a) Collinear spins (b) Non-collinear spins Figure 5.11. Energy change with the angle of collinear and non-collinear spins to [111] direction

5.6 Spin spiral dispersion relation In this section, we ﬁrstly mainly discuss the difference of antiferromagnetic phase and ferromagnetic phase of BiFeO3. In our calculations, the AFM and FM phase have the same crystal structure in order to exclude any unexpected disturbance. The phonon dispersion of the two phases along [111], [110], [110]¯ and [112] are presented in Fig. 5.13. The AFM BiFeO3 obtained from calculation agrees well with the result from inelastic x-ray scattering[52]. The phonon band structure of FM BiFeO3 has imaginary frequency near the gamma point along the [110], [110]¯ and [112] direction, which indicates the instability of its structure.

39 Figure 5.12. (a) The magnetic moments of the two Fe atoms M1 and M2 are opposite to each other and perpendicular to [111] in collinear conﬁguration of spins. (b) In the non-collinear conﬁguration, magnetic moments of Fe atoms are canted. M1 and M2 do not cancel each other completely, the residual total magnetic moment Mr lies in a plane perpendicular to (111) plane.

(a) Phonon dispersion of AFM BiFeO3

(b) Phonon dispersion of FM BiFeO3

Figure 5.13. Energy change with the angle of collinear and non-collinear spins to [111] direction Figure 5.14. Spin spiral dispersion of AFM and FM BiFeO3 along [101]¯ direction

The AFM BiFeO3 is reported to have a spiral spin order with an incom- mensurate long-wavelength of 620 Å[8] along [110] in hexagonal lattice that is equalivent to [101]¯ in rhombohedral lattice. The energy of FM BiFeO3 decreases as the wave vector q increase along the [101]¯ direction and equals to the energy of AFM phase at the q=[0.34 0 -0.34]. The different dispersion re- lations indicate that the the long-wave spiral has opposite effects on the AFM and FM phase.

41 6. Summary and Outlook

In this thesis, the structure, electronic and magnetic properties of BiFeO3 were studied in detials by the density functional theory (DFT). In Chapter 2, 3, 4, the theoretical background related to this study are introduced. Investigating the physical properties of materials is a many body problem that cannot be solved analytically. The DFT simplify the many body problem to the one-particle Kohn-Sham equation, where the total energy E[n] is written as a functional of the electron density n(r). The exact from of the exchange-correlation functional Exc[n] is not given by DFT itself. The local spin density approximation (LSDA) and generalized gradient approximation (GGA) are commonly used from of the exchange-correlation functional. To improve the accuracy, meta-GGA and hybrid functional were introduced in DFT calculation. The on-site Coulomb repulsion was included in the LDA+U method, in which the strong correlation between localized orbitals are described more reasonable. The wave functions are expanded by plane wave basis and the potential in core region is replaced by a pseudopotential to make the DFT calculation practical on computer. The phonon dispersion is derived by solving the dynamic matrix D(k). The modern polarization theory explains the multiple values of electric polarization in inﬁnite system like periodic lattice. The electronic contribution to the polarization is calculated by treating it as a Berry phase of valence wave functions. Heisenberg model of magnetism explains the formation of different magnetic orders. Non-collinear magnetic phenomenon originates from the spin-orbital coupling. The application and results are discussed in Chapter 5. In this study, Ue f f = 2eV was found to be a reasonable value to include the Coulomb correlation effect into the calculations. With this effective U value, the calculated structural parameters for BiFeO3 are in good agreement with experiments and the band structure of both R3c and R3¯c are insulting. The Berry phase calculations pre- 2 dicted a polarization of 103.5 µC/cm for R3c BiFeO3 which agrees well with experimental measured value for single crystal sample. The (111) planes were found to be magnetization easy plane for R3c BiFeO3 for collinear spins. In the self-consistent calculation, the two spins of neighbor Fe atoms were canted away from antiparallel arrangement, the residual total magnetic moment of 47 mµB/unitcell indicates weak ferromagnetism in R3c BiFeO3. The phonon calculations point out the instability of R3c BiFeO3 with ferromagnetic order. In the spin spiral dispersion, it was found that the energy of FM BiFeO3 decreases as the spin wave vector q increases from gamma point to the boundary of the ﬁrst Brillouin zone along [101]¯ direction, whereas the energy of AFM BiFeO3 increases and has the same value as FM at q=(0.34 0 -0.34).

42 Further investigation should be focused on the coupling between the electric polarization and magnetic order. There are three suggested approaches to investigate the coupling: (i) To study the magnetic order under applied electric filed along [111]; (ii) To study the effect of magnetic field on electric polarization; (iii) To study the effect of lattice strain along [111] direction on both magnetic order and polarization. This will make it possible to use electric field to control the magnetic magnetic moments or to use magnetic field to control the polarization, which is essential for magnetoelectric devices.

43 7. Acknowledgements

I would like to thank Biplab Sanyal for his supervision, suggestion and en- thusiasm, and Bo Yang, Do Wang, Raquel Esteban Puyuelo, Xin Chen for their kind assistance and encouragement. I can’t make so much progress in academics and personal abilities in this two years without them. Specially, I want to thank to my wife Xiaoyu Wen for her loving considerations and great conﬁdence in me all through these two years when we were separated in two different countries.

44 References

[1] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev., 136:B864–B871, Nov 1964. [2] Smolenski˘ı. Ferroelectromagnets. [3] Hans Schmid. Multi-ferroic magnetoelectrics. Ferroelectrics, 162(1):317–338, 1994. [4] Nicola A Hill. Why are there so few magnetic ferroelectrics?, 2000. [5] Tsuyoshi Kimura, T Goto, H Shintani, K Ishizaka, T-h Arima, and Y Tokura. Magnetic control of ferroelectric polarization. nature, 426(6962):55, 2003. [6] SV Kiselev. Detection of magnetic order in ferroelectric bifeo_3 by neutron diffraction. Sov. Phys., 7:742, 1963. [7] James R Teague, Robert Gerson, and William Joseph James. Dielectric hysteresis in single crystal bifeo3. Solid State Communications, 8(13):1073–1074, 1970. [8] I Sosnowska. I. sosnowska, t. peterlin-neumaier, and e. steichele, j. phys. c 15, 4835 (1982). J. Phys. C, 15:4835, 1982. [9] T Zhao, A Scholl, F Zavaliche, K Lee, M Barry, A Doran, MP Cruz, YH Chu, C Ederer, NA Spaldin, et al. Electrical control of antiferromagnetic domains in multiferroic bifeo 3 films at room temperature. Nature materials, 5(10):823, 2006. [10] JT Zhang, XM Lu, J Zhou, H Sun, J Su, CC Ju, FZ Huang, and JS Zhu. Origin of magnetic anisotropy and spiral spin order in multiferroic bifeo3. Applied Physics Letters, 100(24):242413, 2012. [11] JBNJ Wang, JB Neaton, H Zheng, V Nagarajan, SB Ogale, B Liu, D Viehland, V Vaithyanathan, DG Schlom, UV Waghmare, et al. Epitaxial bifeo3 multiferroic thin film heterostructures. science, 299(5613):1719–1722, 2003. [12] JB Neaton, C Ederer, UV Waghmare, NA Spaldin, and KM Rabe. First-principles study of spontaneous polarization in multiferroic bi fe o 3. Physical Review B, 71(1):014113, 2005. [13] Aswini K Pradhan, Kai Zhang, D Hunter, JB Dadson, GB Loiutts, P Bhattacharya, R Katiyar, Jun Zhang, David J Sellmyer, UN Roy, et al. Magnetic and electrical properties of single-phase multiferroic bifeo 3. Journal of Applied Physics, 97(9):093903, 2005. [14] YP Wang, GL Yuan, XY Chen, JM Liu, and ZG Liu. Electrical and magnetic properties of single-phased and highly resistive ferroelectromagnet bifeo3 ceramic. Journal of Physics D: Applied Physics, 39(10):2019, 2006. [15] Delphine Lebeugle, Dorothée Colson, A Forget, and Michel Viret. Very large spontaneous electric polarization in bi fe o 3 single crystals at room temperature and its evolution under cycling fields. Applied Physics Letters, 91(2):022907, 2007.

45 [16] P Ravindran, R Vidya, A Kjekshus, H Fjellvåg, and O Eriksson. Theoretical investigation of magnetoelectric behavior in bi fe o 3. Physical Review B, 74(22):224412, 2006. [17] Paul AM Dirac. Note on exchange phenomena in the thomas atom. In Mathematical Proceedings of the Cambridge Philosophical Society, volume 26, pages 376–385. Cambridge University Press, 1930. [18] M Gell-Mann. M. gell-mann and ka brueckner, phys. rev. 106, 364 (1957). Phys. Rev., 106:364, 1957. [19] WJ Carr Jr and AA Maradudin. Ground-state energy of a high-density electron gas. Physical Review, 133(2A):A371, 1964. [20] E Wigner. E. wigner, phys. rev. 46, 1002 (1934). Phys. Rev., 46:1002, 1934. [21] John P Perdew and A Zunger. Self-consistent equations including exchange and correlation effects phys. Rev. B, 23:5048–79, 1981. [22] DM Ceperley. Dm ceperley and bj alder, phys. rev. lett. 45, 566 (1980). Phys. Rev. Lett., 45:566, 1980. [23] Stephen A Gramsch, Ronald E Cohen, and Sergej Yu Savrasov. Structure, metal-insulator transitions, and magnetic properties of feo at high pressures. American Mineralogist, 88(2-3):257–261, 2003. [24] John P Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized gradient approximation made simple. Physical review letters, 77(18):3865, 1996. [25] John P Perdew and Yue Wang. Accurate and simple analytic representation of the electron-gas correlation energy. Physical Review B, 45(23):13244, 1992. [26] Fabien Tran and Peter Blaha. Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Physical review letters, 102(22):226401, 2009. [27] Axel D Becke. Density-functional thermochemistry. iii. the role of exact exchange. The Journal of chemical physics, 98(7):5648–5652, 1993. [28] Axel D Becke. Density-functional exchange-energy approximation with correct asymptotic behavior. Physical review A, 38(6):3098, 1988. [29] Seymour H Vosko, Leslie Wilk, and Marwan Nusair. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Canadian Journal of physics, 58(8):1200–1211, 1980. [30] Chengteh Lee, Weitao Yang, and Robert G Parr. Development of the colle-salvetti correlation-energy formula into a functional of the electron density. Physical review B, 37(2):785, 1988. [31] Vladimir I Anisimov, Ferdi Aryasetiawan, and AI Lichtenstein. First-principles calculations of the electronic structure and spectra of strongly correlated systems: the lda+ u method. Journal of Physics: Condensed Matter, 9(4):767, 1997. [32] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton. Electron-energy-loss spectra and the structural stability of nickel oxide: An lsda+u study. Phys. Rev. B, 57:1505–1509, Jan 1998. [33] D R Hamann, M Schluter, and C Chiang. Norm-conserving pseudopotentials. Physical Review Letters, 43(20):1494–1497, 1979. [34] P. E. Blöchl. Projector augmented-wave method. Phys. Rev. B, 50:17953–17979, Dec 1994. [35] R. Resta. Theory of the electric polarization in crystals. Ferroelectrics,

46 136(1):51–55, 1992. [36] R. D. King-Smith and David Vanderbilt. Theory of polarization of crystalline solids. Phys. Rev. B, 47:1651–1654, Jan 1993. [37] Igor Dzyaloshinsky. A thermodynamic theory of âweakâ ferromagnetism of antiferromagnetics. Journal of Physics and Chemistry of Solids, 4(4):241–255, 1958. [38] Tôru Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Physical Review, 120(1):91, 1960. [39] Claude Ederer and Nicola A. Spaldin. Weak ferromagnetism and magnetoelectric coupling in bismuth ferrite. Phys. Rev. B, 71:060401, Feb 2005. [40] Matthias Menzel, Yuriy Mokrousov, Robert Wieser, Jessica E. Bickel, Elena Vedmedenko, Stefan Blügel, Stefan Heinze, Kirsten von Bergmann, André Kubetzka, and Roland Wiesendanger. Information transfer by vector spin chirality in finite magnetic chains. Phys. Rev. Lett., 108:197204, May 2012. [41] Tôru Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev., 120:91–98, Oct 1960. [42] Peter E Blöchl. Projector augmented-wave method. Physical review B, 50(24):17953, 1994. [43] Georg Kresse and D Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 59(3):1758, 1999. [44] Georg Kresse and Jürgen Furthmüller. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Physical review B, 54(16):11169, 1996. [45] Georg Kresse and Jürgen Hafner. Ab initio molecular dynamics for liquid metals. Physical Review B, 47(1):558, 1993. [46] GP Vorob’ev, AK Zvezdin, AM Kadomtseva, Yu F Popov, VA Murashov, and DN Rakov. Possible coexistence of weak ferromagnetism and spatially modulated spin structure in ferroelectrics. Physics of the Solid State, 37:1793–1795, 1995. [47] Hendric J Monkhorst. Hj monkhorst and jd pack, phys. rev. b 13, 5188 (1976). Phys. Rev. B, 13:5188, 1976. [48] Frank Kubel and Hans Schmid. Structure of a ferroelectric and ferroelastic monodomain crystal of the perovskite bifeo3. Acta Crystallographica Section B: Structural Science, 46(6):698–702, 1990. [49] Ram Seshadri and Nicola A Hill. Visualizing the role of bi 6s âlone pairsâ in the off-center distortion in ferromagnetic bimno3. Chemistry of materials, 13(9):2892–2899, 2001. [50] David Vanderbilt and RD King-Smith. Electric polarization as a bulk quantity and its relation to surface charge. Physical Review B, 48(7):4442, 1993. [51] Raffaele Resta. Macroscopic polarization in crystalline dielectrics: the geometric phase approach. Reviews of modern physics, 66(3):899, 1994. [52] Elena Borissenko, Marco Goffinet, Alexei Bosak, Pauline Rovillain, Maximilien Cazayous, Dorothée Colson, Philippe Ghosez, and Michael Krisch. Lattice dynamics of multiferroic bifeo3 studied by inelastic x-ray scattering. Journal of Physics: Condensed Matter, 25(10):102201, 2013. [53] Jiefang Li, Junling Wang, Manfred Wuttig, R Ramesh, Naigang Wang, B Ruette, Alexander P Pyatakov, AK Zvezdin, and D Viehland. Dramatically

47 enhanced polarization in (001),(101), and (111) bifeo 3 thin ﬁlms due to epitiaxial-induced transitions. Applied physics letters, 84(25):5261–5263, 2004. [54] A Togo and I Tanaka. First principles phonon calculations in materials science. Scr. Mater., 108:1–5, Nov 2015. [55] I. Dzyaloshinsky. A thermodynamic theory of "weak" ferromagnetism of antiferromagnetics. Journal of Physics and Chemistry of Solids, 4(4):241 – 255, 1958.