Examensarbete 30 hpOktober 2018

First Principle Study of Multiferroic Bismuth Ferrite

Yuhang Liu

Masterprogrammet i fysikMaster Programme in Physics

AbstractIn this work, the spontaneous magnetic and electric behavior of the multiferroic R3c BiFeO3 arestudied 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 inequilibrium. The elongation of perovskitelike lattice along the [111] direction is correctly pre-dicted by GGA+U method. We predicated a large electric polarization of 103.5 µC/cm2 byberry phase method, which is consistent with the experimental measured polarization of 100µC/cm2 for high quality single crystal sample. Our calculations indicate that the (111) planesare magnetic easy planes and the magnetic anisotropy origins from both single-ion anisotropyand the superexchange coupling. With the superexchange included in our calculations, theground state of BiFeO3 is found to have an approximate antiferromagnetic (AFM) order withweak ferromagnetic (FM). Our Phonon dispersion analysis shows the instability of the hypo-thetical FM phase of BiFeO3. We found the the energy of AFM and FM BiFeO3 evolve inopposite ways in spin-sprial dispersion and equals to each other at q=[0.34 0 -0.34].

Dedicated to all hard-workingresearchers at Uppsala University

Contents

1 Introduction to this study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Populävetenskaplig Sammanfattning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1 Many Body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Kohn-Sham equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Exchange correlation functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Local spin density approximation (LSDA) . . . . . . . . . . . . . . . . . . . . 152.3.2 Generalized gradient approximation (GGAs) . . . . . . . . . . . . . . . . 162.3.3 Beyond GGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.4 L(S)DA+U: On-site Coulomb-repulsion . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4 Plane wave sets and pseudopotential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 First principles phonon calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Modern polarization theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.1 Polarization lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Berry phase theory for the macroscopic polarization . . . . . . . . . . . . . . . . . 24

4 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.1 Heisenberg Hamiltonian and Magnetic ordering . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.2.1 Single-ion anisotropy and spin-orbital coupling . . . . . . . . . . . 274.2.2 Non-collinear magnetism and Dzyaloshinskii-Moriya

interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

5 Results and discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.1 Computational details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.2 Structure of BiFeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.3 Electronic properties of BiFeO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335.4 Electric polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.5 Magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.6 Spin spiral dispersion relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

6 Summary and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

7 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

1. Introduction to this study

1.1 IntroductionIt is safe to say that the progress in science and technology is mainly focusedon the understanding and application of materials. To comprehend a propertyof a material, the most direct and effective way is testing it by experiments.Every advancement in testing technology brings huge a progress in exploringmaterials, for instance, the invention of X-ray enabled researchers to study theperiodic structure of crystal and the scanning tunneling microscope expandedthe human’s observation to the scale of molecule and atom. However, themore advanced technology tends to consume more resource. The increasingcost and difficulty of experimental observation have made the experiments nolonger the most efficient approach to study materials and even sometimes it isno longer possible. For example, to study the phase transition under ultra-highpressure, creating the high pressure in laboratory is not a wise choice and evenworse, sometimes current technology cannot reach the high pressure needed.Another specific example is that people cannot and don’t need to create a starif 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 thefundamentals of nature, whereas the philosophy of theory is totally differentfrom experiment. Through an experiment, the phenomenon is always knownbut the origins of the phenomenon are not, and researchers obtain data fromexperiments and look for some theories to explain them. . If no theory canexplain it, experimentalists will claim that current theories fail but a new phe-nomenon is found. For theory, researchers make assumptions and build up atheory based on these assumptions. If the theory agrees with experiments, itis 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 isnecessary to emphasize that theory can describe how the nature behaves butcannot always explain why the nature behaves like that, and theory can neverbe proved to be correct but can be proved to be wrong by experiments. Peoplemay wonder why we should still trust in a theory if it cannot be proved to becorrect. My answer to this question is that if a theory has been proven suffi-ciently appropriate for a phenomenon during a scientific experiment then weshould think that the theory will also be a suitable and credible explanationagain in the future. In general, a theory is a model we build based on knownphenomena through experiments and an appropriate theory is a model that

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works every time and can be a contributing factor to our future experimentswhen we are explaining and analyzing unknown phenomena.

At this point, I believe we should have a relatively positive and objectiveunderstanding of the theory. The quantum mechanics has been proved reliablein describing electronic system, so it should be able to calculate the propertiesof materials which are made up of electrons and ions. In principle, a finite orinfinite electron-ion system can be described by a many-body wave functionno matter how complex the system is, and if the many-body wave functionis solved, all properties of this material will be reachable. However, solv-ing such a many-body wave function for a macroscopic number of atoms isimpossible for human even with the most advanced computer. For most prop-erties, solving the many-body wave function is sufficient but not necessary.Density functional theory (DFT) transforms the many-body wave function toa single particle Kohn-Sham function and the total energy is expressed as afunctional of the electron density function but not a function of the positionsof all electrons[1].

Multiferroics[2][3] are materials with two or more ferroic order parameterssimultaneously in the same place. The coupling between two order parame-ters provides novel properties and make multiferroics special. For instance,coupling between the spontaneous polarization of ferroelectricity and spon-taneous magnetization of ferromagnetism results in magnetoelectric (ME) ef-fects, in which the magnetization can be tuned by electric field and the electricpolarization can be tuned by magnetic field. The magnetoelecrics with MEeffects are of great interest for potential application in nondestructive datareading and writing with low energy consumption. Besides the prospectiveapplications, the physics behind the magnetoelectric materials is still interest-ing and attractive. The magnetoelectrics are not common but actually veryrare in nature. The reason is that the ions tend to be magnetic, which usu-ally tend to the absence of electric polarization. The non-zero total magneticmoment requires partially filled 3d orbitals in transition metals or 4f orbitalsin lanthanides. Whereas 3d electrons of transition metals tend to prohibit theferroelectric distortion[4]. It has been shown in TbMnO3 that electric plar-ization can be controled by the external magnetic field[5], whereas its smallpolarization 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 antifer-romagnetic with a high Neél temperature of 643 K[6][7]. The structure ofBiFeO3 is a highly distorted rhombohedral perovskite structure with R3c spacegroup as shown in Fig.1.1. The distortion consists of two parts: (i) counter-rotations of neighboring O6 octahedra about the [111] direction. (ii) anionsand cations displacement along [111] direction. BiFeO3 has a G-type anti-ferromagnetic order as shown in Fig.1.2, in which Fe magnetic moments arecoupled ferromagnetically within the (111) planes and antiferromagneticallybetween neighbor (111) planes. In bulk BiFeO3, the magnetic order is modi-

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fied by a spiral spin order in which antiferromagnetic vector rotates with a longwave-length of 620 Å[8]. The propagation direction is [111] in hexagonal lat-tice and [110] in rhombohedral lattice. The spin spiral order is suppressed inthin film BiFeO3[9]. Theoretical analysis showed that the spin spiral struc-ture originates from the competition between the isotropic superexchange andthe Dzyaloshinskii-Moriya interaction of the nearest neighbor spins of Featoms[10]. In thin films at room temperature, electric polarization of 50-60µC/cm2 has been measured in experiments[11]. Theoretical studies usingDFT 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 beenmeasured[13][14], which is in sharp contrast with the large atomic displace-ments of BiFeO3. A very large spontaneous electric polarization above 100µC/cm2 was measured in single crystals of BiFeO3 synthesized by a fluxgrowth method[15]. The hybridization interaction between Bi-O and Fe-Ohas been found playing important role in the ferroelectric polarization of R3cBiFeO3 in theoretical study[16].

The purpose of this study is to calculated the electric polarization and mag-netic order and to understand the origin of electric and magnetic propertiesof BiFeO3. At the beginning, the ground state crystal structure of BiFeO3 ispredicted by DFT with GGA+U method. The most appropriate U value is de-termined by comparing the calculated structural parameters and experimentalresults. The spontaneous polarization is calculated along a assumed transitionpath by the berry phase method. To study the origin of magnetic order andelectric polarization, the centrosymmetric R3c phase of BiFeO3 is also stud-ied by DFT and compared with the R3c BiFeO3. The more accurate magneticorder with canted magnetic moments is calculated in self-consistent calcula-tions in which the superexchange is included. To understand the difference ofthe ground state AFM phase and the hypothetical FM phase of BiFeO3, thephonon dispersion and spin spiral dispersion are calculated.

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

Figure 1.2. G type antiferromagnetic order

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1.2 Populävetenskaplig SammanfattningFramför allt kan man säga att framstegen inom vetenskap och teknik fokuserarmer påförstoch tillämpning av material. För att förståen egenskap av ett mate-rial mett direkt och effektivt experiment utföras. Varje framsteg inom provn-ingsteknik ger stora framsteg till materialforskning, till exempel röntgenteknikoch kristallografi. Genom Röntgenteknik och kristallografi har forskare mermöjligheter att ta reda påden periodiska kristallstrukturen medan sveptun-nelmikroskopet expanderade människans iakttagelser till en molekylär ochatomär nivå. Emellertid föreligger en tendens att de avancerade teknik kom-mer att konsumera mer resurser. ökade kostnader och svi experimentobserva-tion har gjort experimentet inte längre det mest effektiva sättet att studera ochforska. I vissa fall är det omöjligt att utföra experiment längre. Till exempelatt det inte är ett klokt val att skapa ett högtryck i laboratoriet för att stud-era fasövergunder ultrahögtryck och ännu värre, ibland kan den nuvarandetekniken inte nådet höga trycket som behövs. Ett annat specifikt exempel äratt människor inte kan och behöver inte skapa en stjärna om de vill undersökasolen.

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 fenomenensursprung är dock svatt veta. Brukar fforskare data frett experiment och deletar efter nteorier som kan vara en lämplig förklaring till fenomenen samtsina upphov. Om det inte finns nteori kan förklara upphovsorsaker, dåhävdarexperimentalister att teorier misslyckas och ett nytt fenomen hittas. För teorigör forskare antaganden och bygger upp en teori utifrdessa antaganden. Omteorin överensstämmer med experiment, dåär den en bra teori. Om teorin intekan förklara vad som finns i experiment, kommer teoretiker att hävda att teorinföreligger en brist och behöver alltsåförbättras. Det är nödvändigt att betonaatt olika teorier kan beskriva hur naturen beter sig men de kan inte förklaravarför naturen beter sig så. Pådet andra sätet kan teorier aldrig bevisas varakorrekta men kan motbevisas vara fel genom experiment. Människor kanskeundra varför vi fortfarande sätta vlit till en teori om det inte kan bevisas varakorrekt. Mitt svar påden här frär att om en teori har bevisats tillräckligt lämpligför ett fenomen under ett vetenskapligt experiment dåbör vi tro att teorin ävenkommer att vara en lämplig och trobar förklaringar igen i framtiden. I generellär teorier en modell som vi har byggt utifrkända fenomen genom experiment.Lämpliga teorier är ocksåen modell som kan vara en bidragande faktor tillvkommande experiment när vi ska förklara och analysera okända fenomen iframtiden. Vid det här laget tror jag att vi bör ha en relativt positiv och ob-jektiv förstav teorier. Kvanteringsmekaniken har visat sig vara tillförlitlig vidbeskrivningen av elektroniskt system och därför har vi möjlighet att beräknaolika egenskaper hos material som tillverkas av elektroner och joner. I prin-

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cip kan en Schrodinger-ekvation och alla andra partiklar i systemet ge oss allainformation om det här systemet men det är bara en princip. I praktiken kom-mer 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 intenödvändigt att lösa en såstor Schrodinger-ekvation om vi vill undersöka enegenskap. Täthetsfunktionsteori (DFT) ger oss ett sätt att kosta mindre ochfånästan samma. DFT är inte helt korrekt eftersom vissa nödvändiga fören-klingar ingi praktiska beräkningar. DFT kan ge oss goda förutsägelser omegenskaper sstruktur, magnetism och spektrum inom msystem. DFT utveck-lar fortfarande mmetoder och de kommer att göra DFT kraftfullare och merexakt.

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 förekommai andra icke-fero material. Exempelvis innehmagnet en ferromagnetisk egen-skap eftersom den visar magnetism i frav yttre magnetfält, medan en aluminiu-minte innehen ferromagnetisk egenskap eftersom den endast visar magnetismom den befinner sig i ett yttre magnetfält. Multiferroic betyder att ett materialsom har tvåeller flera egenskaper som verkar pådet ferroiska sättet. BiFeO3har bferromagnetisk och ferroelektrisk egenskap. BiFeO3 har icke-noll elek-trisk polarisering i frav ett externt elektriskt fält samt ett icke-nollmagnetisktmoment i frav ett externt magnetfält. Multiferroics är intressant, eftersomsamexistensen av tvåferroparametrar inte är vanligt och samspelet mellan detvåparametrarna 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, elektriskpolarisering och magnetisk egenskap hos BiFeO3.

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2. Density Functional Theory

2.1 Many Body problemThe main purpose of condensed matter physics is to describe the behaviorof electrons in atoms, molecules and solids. The most fundamental equationdescribing 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 thetotal energy of the system. H is the Hamiltonian which contains kinetic termsand interaction energy terms of both nuclei and electrons, it can be expressedas

H =−∑I

h2

2mI∇

2I +

12 ∑

I 6=J

ZIZJe2

|RI−RJ|−∑

i

h2

2me∇

2i +

12 ∑

i6= j

e2

|ri− r j|−∑

i,I

ZIe2

|RI− ri|(2.2)

where RI , ri represent the positions of nuclei and electrons, mI and mI repre-sent the mass of the nuclei and electrons respectively. ZI is the atomic num-ber.In this Hamiltonian, the first term is the kinetic energy of all nuclei, thesecond term is the Coulomb interaction between nuclei, the third term is thekinetic energy of all electrons, the fourth term is the Coulomb interaction be-tween electrons and the last term is the Coulomb interaction between electronsand nuclei. Solving the exact solution of the Hamiltonian of all nuclei andelectrons is almost impossible. The most simple but powerful approximationto be made is the Born-Oppenheimer Approximation. The mass of electronis only 1/1836 of that of proton, so they are much lighter than nuclei andmove much faster than nuclei. Therefore the electrons respond almost instan-taneously to any movement of nuclei, the nuclei are almost static comparedto electrons. This approximation makes it possible to separate the nuclei partfrom the electronic part in the Hamiltonian. The Hamiltonian of electrons canbe expressed as,

H =−∑i

h2

2me∇

2i +

12 ∑

i6= j

e2

|ri− r j|−∑

i,I

ZIe2

|RI− ri|(2.3)

The problem is simplified to work out the electronic wave function ψ(R,r)which is governed by the kinetic energy, electron-electron Coulomb interac-tion and the static Coulomb potential field of nuclei. However, this Schrödingerequation is still unsolvable due to the complexity of the electron-electronCoulomb interaction. Thus a better method is needed to describe electrons.

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2.2 Kohn-Sham equationAccording to Hohenberg-Kohn theorems[1], for an interacting electrons in anexternal potential Vext

Theorem 1 The external potential Vext can be determined by the groundstate density n0(r) except for a constant.

Theorem 2 The density n(r) that minimizes the total energy functionalE[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 vari-ational principle. Following their approach, we can write the total energyfunctional E[n] as

E[n] = T [n]+∫

Vext(r)n(r)dr+12

∫ e2n(r)n(r′)|r− r′|

drdr′+Exc[n] (2.4)

in which T[n] is the kinetic energy functional of the hypothetical non-interactingelectrons, Vext is the external potential due to the nuclei or ions, the third termis the Coulomb (Hartree) energy, and Exc[n] is the exchange-correlation en-ergy which includes all many-body effects. The exchange-correlation part ofkinetic energy Txc is included in Exc We can use the variational principle onthe total energy functional E[n] and the minimization of the energy functionalresults in the Kohn-Sham(KS) equation,

[− h2

2m∇

2 +Ve f f (r)]φi(r) = εiφi(r) (2.5)

where the effective potential Ve f f is defined as

Ve f f (r) =Vext(r)+∫ e2n(r′)|r− r′|

dr′+Vxc(r) (2.6)

where the exchange-correlation potential is expressed as

Vxc =δExc[n]δn(r)

(2.7)

The density is

n(r) =N

∑i|φi(r)|2 (2.8)

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

N

∑i

εi = E +12

∫ e2n(r)n(r′)|r− r′|

drdr′−Exc[n]+∫

δExc[n]δ [n(r)]

n(r)dr (2.9)

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2.3 Exchange correlation functionalsThe DFT theory successfully separates the single particle kinetic energy andthe Coulomb (Hartree) energy from the many-body exchange-correlation func-tional, which can be expressed in many different approaches, such as the localspin density approximation (LSDA) and the generalized gradient approxima-tions (GGA).

2.3.1 Local spin density approximation (LSDA)Hohenberg and Kohn already suggested the local density approximations intheir first DFT paper[1]. They pointed out that electrons in solids can be of-ten considered as homogeneous electron gas. The LDA exchange-correlationfunctional has a quite simple form.

ELDAxc [n(r)] =

∫n(r)εxc(n(r))dr (2.10)

Where εxc(n) represents the exchange-correlation energy density of a homo-geneous electron gas with density n(r).

The local spin-density approximation (LSDA) is a generalization of LDAformulated in terms of two spin densities n ↑ (r) and n ↓ (r).

Exc[n↑,n↓] =∫

n(r)εxc(n↑,n↓)dr (2.11)

The exchange-correlation energy Exc can be decomposed into exchange andcorrelation 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 fromthe Dirac’s work in 1930[17].

Ex[n(r)] =−k∫

n43 (r)dr (2.13)

where k = 32(

34π)

13 for LSDA and k = 2−

13 3

2(3

4π)

13 for LDA.

Unfortunately, we only know analytic expressions for the correlation partEcin the high[18][19] and low[20] density limits. A commonly used from isthe interpolation formula of Perdew and Zunger [21] of which interpolationcoefficients are derived from the date of quantum Monte Carlo of the homo-geneous electron gas generated by Ceperley and Alder[22].

Despite LDA’s simplicity, it gives good predictions for system with slowlyvarying charge densities. LDA’s prediction of lattice constants is accurate towithin a few percent. But LDA has several deficiencies, it tends to give higherbinding energy. In magnetic system, LDA may gives wrong prediction of mag-netic order, for example Fe is predicted to be FCC paramagnetic by LDA, but

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it has bcc ferromagnetic order in experiment. In strongly correlated systemswhere particles interact with each other, LDA gives inaccurate result. For in-stance, LDA predicts transition metal oxides FeO,CoO, NiO and MnO to bea metals or semiconductors, but they are all Mott insulators. With the help ofHubbard-corrected functionals, the LDA+U method which will be discussedin 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. Animprovement can be easily considered is to include the gradient of the electrondensity, then we have the generalized gradient approximations (GGA). Thegeneral form of GGA is:

EGGAxc [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 areconstants, it is a simplification of the PW91[25]. Comparing with LDA, GGAscorrect the overestimated binding energy , give correct prediction for magneticsystems such as Fe’s BCC ferromagnetic order and perform better for bulkphase stability ect.. However, similar to LDA, GGAs are inaccurate to describeband gap of transition metal and rare earth compounds.

2.3.3 Beyond GGAIn meta-GGA, the second order gradient, Laplacian, is included in the func-tional. The Laplacian ∇2n can be considered as the effect of the kinetic energyof electrons’ movement. It has be shown that accurate band gaps could be ob-tained for many materials with meta-GGA[26].

Some functionals always overestimate energy some others always under-estimate energy, people may ask what if we use the combination of func-tionals. That is the idea of hybrid functionals. Most hybrid functionals takepart of Hartree-Fock energy, LDA energy and GGA energy into the exchange-correlation energy, for example the most successful functional for chemistryB3LYP [27]:

EB3LY Px = 0.8ELDA

x +0.2EHFx +0.72∇EB88

x (2.15)

EB3LY Pc = 0.19EV MN3

c +0.81ELY Pc (2.16)

Where EB88x is the exchange part of the Becke 88 exchange functional[28], the

EV MN3c is the correlation energy of Vosko-Wilk-Nusair correlation functional

16

III[29] and ELY Pc 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- orGGA-type calculations coupled with an additional" orbital-dependent inter-action. The strong on-site Coulomb interaction of localized electrons is notcorrectly 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 byintroducing the Coulomb interaction U into the LDA energy functional. In ourcalculations, a simplified form of LDA+U introduced by Dudarev[32] et al isused. In Dudarev’s form, the energy functional is of the form:

ELDA+U = ELDA +U− J

2 ∑[(∑m1

nσm1,m1)− ( ∑

m1,m2)nσ

m1,m2nσm2,m1] (2.17)

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 pseudopotentialIn 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. Anyperiodic function can be expanded with a discrete set of plan wave functionsby Fourier transformation method. Thus all wave functions can be expandedwith a discrete plan-wave basis set in periodic lattice. Only a basis set withinfinite functions can expand a wave function completely. However, a finitebasis set with a suitable energy cutoff is enough to expand a wave functionprecisely. That is because functions with lower energy (lower frequency) areof more importance than functions with higher energy (higher frequency). Theabsence of high frequency part will cause error in total energy, thus in calcu-lation the energy cutoff should be high enough to make the error smaller thanthe tolerance.

Since the wave function in core region oscillate rapidly, a plane-wave ba-sis set with finite cutoff energy usually fails in expanding core orbitals andwave function of valence electrons in the core region. In the pseudopoten-tial approximation[33], a real atomic potential is replaced by a more smoothpseudopotential which make it easier to be expanded by plan-wave basis.

The valence electrons are more essential to physical properties of solid crys-tals than core electrons, since chemical bonding happens in the inter-core re-gion. 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 isdefined as r ≤ rc

.

and is identical to atomic potential out of the core region, which makes thepseudo wave function nodeless in core region and easier to be expanded byplan-wave basis. A schematic representation of pseudopotential is shown inFig. 2.4.

The pseudopotential introduced by Hamman, Schluter and Chiang was con-structed from norm conserving pseudo potentials (NCPP). The nodeless pseudo-wavefunction ψpseduo conserves norm and energy eigenvalues of the true allelectrons wave function. The NCPP gives accurate results but still requiresmuch 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 pseu-dopotential method. The main goal of this method is to avoid using the os-cillatory true atomic wave functions when calculating physical properties likeexpectation values and densities. The APW method starts from expanding antrue oscillatory all-electrons single particle KS wave function ψn with a set ofsmooth pseudo wave functions ψn by a linear transformation T :

|ψn〉= T |ψn〉 (2.19)

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

18

form:T = 1+∑

aTa (2.20)

in order to make the all-electrons wave function and smooth pseudo wavefunction identical out the core region. In another word, T only transform wavefunction in the core region. Each core region is defined in a augmentationsphere with cut-off radius ra

c around an atom as:

|r−Ra| ≤ rac (2.21)

The solutions to the Kohn-Sham Schrodinger equation for an isolated atomare partial waves which also form a complete basis. T transforms the partialwaves φi to the smooth partials waves φi.

|φ ai 〉= T

∣∣φ ai

⟩(2.22)

In augmentation spheres the smooth pseudo wave function is expanded intosmooth partial waves:

|ψn〉= ∑i

Cani∣∣φ a

i⟩

(2.23)

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 wavesφ a

i by the same parameters Cani.

T |ψn〉= T ∑i

Cani∣∣φ a

i⟩

(2.24)

T |ψn〉= ∑i

CaniT∣∣φ a

i⟩

(2.25)

|ψn〉= ∑i

Cani |φ a

i 〉 (2.26)

The expanding coefficients Cani can be expressed as the projection of smooth

wave function ψn onto projector functions pi.

Cani =

⟨pa

i

∣∣ ˜ψnx⟩

(2.27)

The projector functions pai are of completeness and orthogonality in the aug-

mentation spheres.∑

i

∣∣φ ai⟩⟨

pai

∣∣= 1 (2.28)

⟨pa

i

∣∣φ ai⟩= δi, j (2.29)

By using the completeness and orthogonality, we can derive the expression ofT:

T = 1+∑a

∑i(|φ a

i 〉−∣∣φ a

i

⟩)〈pa

i | (2.30)

19

The transformation T is defined by all-electrons partial waves φ ai , smooth

pseudo partial waves φ ai and projector functions pa

i .The expectation value of any operator O is calculated by the inner product:

oi = 〈ψ| O |ψ〉 (2.31)

The expectation oi can be expressed as the inner product of smooth pseudowave function and operator.

oi = 〈ψ|T †OT |ψ〉 (2.32)

where T †OT can be defined as a smooth pseudo operator ˜O. The frozen coreapproximation is always used with the PAW method. In this approximation,the core states of the isolated atoms are assumed not affected by the forma-tion of chemical bonds. Only valence electrons orbitals are considered in thisapproximation, which reduce the computation cost significantly.

2.5 First principles phonon calculationsThe harmonic approximation is commonly used in phonon frequency calcula-tion. Atoms are assumed oscillating around their equilibrium positions r withdisplacements d. The potential energy function Φ is assumed to be a functionof the displacements around equilibrium positions up to the second order.Thepotential is expressed in the following series form:

Φ = Φ0 + ∑a=x,y,z

∑ln

Φa(ln)da(ln)+12! ∑

a,b∑

ln,ln′Φab(ln, l′n′)da(ln)db(l′n′)

(2.33)where a, b=x, y, z represent directions of three Cartesian coordinate axes and land n are the labels of unit cell and atom in the unit cell respectively.Φ0 is thezeroth order force constants which does not depend on the positions of atoms,it is set to be 0 usually. Φa(ln) is the first order force constant acting on the nth

atom in the lth unit cell along a direction which only depends on the position ofone atom and one direction.Φa,b(ln, l′n′) is the more complicate second orderforce 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 direc-tions at constant volume are calculated by first principle method such as DFTmethod. A force Fa(ln)and a second-order constant Φa,b(ln, l′n′) are obtainedby partial derivatives of potential energy.

Fa(ln) =−−∂Φ

∂da(ln)(2.34)

Φa,b(ln, l′n′) =∂ 2Φ

∂da(ln)db(l′n′)=−∂Fb(l′n′)

∂da(ln)(2.35)

20

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

Fa(ln) =V −V da(ln)

da(ln)(2.36)

and the second-order derivative is also replaced by

Φa,b(ln, l′n′) =Fb(l′n′)−Fda(ln)

b (l′n′)da(ln)

(2.37)

where the superscript da(ln) means the nth atom is displaced d along the adirection. The force on atoms at equilibrium positions are all zero. With thesesecond-order force constants the dynamical matrix D(q) is constructed as

Dab,nn′(k) = ∑l′

Φa,b(0n, l′n′)2√

mnmn′eik[r(l′n′)−r(0n)] (2.38)

where the .The dynamical matrix describes interaction between the nth atomwith a mass of mnin one unit cell and the n′th atom with a mass of mn′ in allunit cells. The sum of the unit cells usually runs over the nearest neighbors inpractical calculation in order to reduce the cost.

The phonon frequency ωk and polarization vector pk are obtained by solv-ing eigenvalue equation of dynamical matrix D(k),

∑bn′

Dab,nn′(k)pk j,bn′ = ω2k jpk j,an (2.39)

where j is the index of phonon band. The displacement vector of the nth atomin the lth unit cell can be derived from its corresponding polarization vector,

d(ln) =A√

mnpk j,neiqr(ln) (2.40)

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 kpoints connecting high-symmetry points in reciprocal space. The potentialenergy Φ is at its minimum if the crystal is in ground state, which meansany displacement of atom from equilibrium position increase the energy. Thephonon frequencies of a stable phase are real and positive at all k points. Theimaginary frequencies (always shown as negative frequencies in dispersioncurve) means the current unit cell is not the ground state and has a trend ofphase transition.

21

3. Modern polarization theory

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

p = ∑n

qnrn (3.1)

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:

p =∫

q(r)rd3r (3.2)

where r is the position vector relative to the reference point of the system andq(r) is charge density at r. In microscopic and finite system such as moleculeand cluster, the above definition 3.2 works perfectly, different reference pointswill always give the same result. As shown in Fig.3.1, a chain with the positiveand negative charges spaced a distance a/2 apart and so the length of unit cellis a. This chain has inversion symmetry to point charge in it so this chainsystem is non-polar and with a dipole moment density of zero.

In macroscopic scale, the dipole moment density is used to describe thepolarization. For crystal materials, the dipole moment density can be definedas the dipole moment per unit cell. However, the choice of unit becomes atrouble when we are try to use the formula 3.2 to calculate the dipole momentdensity even for the most simple case, one-dimension chain of alternative pos-itive 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 anotherone the negative one sits right to the positive one. Now,let us calculate thedipole moment per unit cell, the reference point is chosen as the center of unitcell. 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 chargeand a farther negative charge but not the neighbor one. For this chain thesevalues are q(1+ n)/2, n is an arbitrary integer. This set of dipole momentdensity values are called polarization lattice. For non-polar system, the valuesin the lattice should be symmetric to the origin point but the origin itself is notnecessary to be in the lattice. In solid state crystal structure , if we let anionscombined with electrons of other anions as dipoles we will also get a polariza-tion lattice. The lattice constant of polarization lattice is called polarizationquantum.

Then let us calculated polarization again for a polar chain in which all pos-itive charges moves distance d relative to the negative charges as shown inFig.3.1. In this distorted chain, we will find out our left part is different withthe right part if we sit on a charge, so this system must be polar and shouldhave an non-zero value of dipole moment density. the dipole moment densityfor 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 polarizationhas no physical meaning, the change in polarization between non-polar andpolar system is the value we can measure in experiments.

3.2 Berry phase theory for the macroscopic polarizationThe electronic contribution to the difference in polarization Pel can be treatedas a berry phase of the valence wave function[35][36]. The change in polar-ization along a transformation pass from non-polar structure to polar structureis

∆Pe = Ppolare −Pnon−polar

e (3.15)

with

Pe =−i f |e|Ω0

N

∑n=1

∫BZ〈unk|∇k |unk〉d3k (3.16)

where Ω0 is the volume of unit cell, f is the occupation number of valencestates, N is the number of occupied bands, unk is the cell-periodic part of theBloch function ψnk. It’s more understandable in the form of localized Wannierfunctions of occupied bands

Pe =−f |e|Ω0

N

∑n=1〈Wn|r |Wn〉 (3.17)

Where Wn is the Wannier function associated with band n. The Wannier centeris defined as

rn = 〈Wn|r |Wn〉 (3.18)

The Wannier center is the average position of the electrons in the Wannierfunction.

It is noteworthy that polarization values obtained from calculations will siton several branches spaced by polarization quantum, only the change of po-larization on the same branch is the true value of polarization.

24

4. Magnetism

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

H =−2J12~S1 · ~S2 (4.1)

where J12 is the exchange constant between the ~S1 and ~S2, J12 = J21 . If J12 >0, the triplet state

|1,1〉= ↑↑ (4.2)

|1,0〉= (↑↓+ ↓↑)/√

2 (4.3)|1,−1〉= ↓↓ (4.4)

with a symmetric spin function and an antisymmetric spatial function haslower energy than the single state

|0,0〉= (↑↓ − ↓↑)/√

2 (4.5)

with a antisymmetric spin part and an symmetric spatial part. So the twoelectrons system has a total spin moment S=1. If J12 < 0, the single state isground state. So the system has a total spin moment S = 0.

In solids, the Heisenberg Hamiltonian is expressed as the sum of exchangeHamiltonians of all pairs of atoms:

H =12(−2∑

i, jJi j~Si · ~S j) =−∑

i, jJi j~Si · ~S j (4.6)

The 1/2 factor is introduced because the exchange interaction between anypair is counted twice in the sum. Without external magnetic field, if Ji j ispositive, parallel arrangement of atomic spins has lower energy and the mag-netic ordering is ferromagnetic. Without external magnetic field, if Ji j < 0,antiparallel arrangement of spins has lower energy and the magnetic order-ing is antiferromagnetic. It is convenient to describe antiferromagnet by theconcept of sublattice. An antiferromagnetic lattice can be divided into onesublattice with all spin up atoms and one sublattice with all spin down atoms.The magnetic moments of the two sublattice in antiferromagnet have oppositedirections and same magnitude. In ferrimagnet, the two magnetic moments ofthe 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 ferrimagnetis shown in fig.4.1 The total magnetic moments of ferromagnet and ferrimag-net are non-zero, the total magnetic moment of antiferromagnet is zero. Thespontaneous magnetic orderings discussed above will be broken by thermaldisturbance, the Currier temperature TC is the temperature above which cer-tain materials lose their permanent magnetic ordering. All magnetic orderingswill transform to paramagnetic phase above TC. The total magnetic moment ofparamagnet depends on the applied external magnetic field as shown in Fig.4.1The paramagnet shows zero magnetic moment in the absence of applied mag-netic field. In the presence of applied magnetic field, paramagnet has the samemagnetic moment direction as the external magnetic field.

4.2 Magnetic anisotropyA system conserving its total energy under the change of magnetization di-rection is magnetic isotropic, otherwise it is magnetic anisotropic. The mag-netic anisotropy energy(MAE) is defined as the energy difference between theground state and energetically unfavorable configuration commonly and it’sused to evaluate how the intensity of anisotropy. The magnetic anisotropy isa result of the breaking of the rotational symmetry, which can be caused bydipole-dipole interaction and spin-orbital coupling. The spin-orbital couplingyields the magnetocrystalline anisotropy, in which we are interested.

26

4.2.1 Single-ion anisotropy and spin-orbital couplingIn quantum field theory, the spin is given naturally from the Lorentz invari-ance. The wave equation of relativistic electron is the Dirac equation. Inthe non-relativistic approximation ( v

c 1), the Dirac equation is reduced to aSchrödinger equation with a revised Hamiltonian,

H =p2

2m− eΦ+

eS ·Bm− p4

8c2m3 −eh2

8c2m2 ∇2Φ− e

2m2c2 S · (∇Φ×p) (4.7)

where the last term is spin dependent. The potential function Φ is approxi-mately spherically symmetric, so the its gradient is pointed to the radical di-rection.

∇Φ =rr

dΦ

dr(4.8)

The the spin dependent term Hsoc takes the following form,

Hsoc =−e

2c2m2rdΦ

drS · (r×p) = ξ L ·S (4.9)

where ξ is the spin-orbital coupling constant which depends on the type ofatoms. The spin angular moment is coupled to the crystal lattice via the orbitalangular moment.

The spin-orbital coupling can be also described in a classical picture. For anparticle in a potential field, the spin-orbital coupling is the relativistic interac-tion between the magnetic moment µ associated with its spin and the magneticfield H generated by the orbital motion of itself in the potential. The magneticfield always tends to align the spins to its own direction, then the free energyof the crystal depends on the angle between he magnetic moment µ and themagnetic field H.

4.2.2 Non-collinear magnetism and Dzyaloshinskii-Moriyainteraction

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

HDM = ∑i6= j

Di j · (S1×S2) (4.10)

where Di j is the DM vector. Unlike the Heisenberg exchange interaction theDM interaction is antisymmetric, the swap of the two spins change the signof 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. TheDM interaction makes the spins favor canted configuration, which can causeweak spontaneous magnetization in anti-ferromagnetic material[39] and spin-spirals[40].

The microscopic mechanism of DM interaction is identified as spin-orbitalcoupling by Moriya[41]. The DM interaction between two neighbor spins istransferred by a third atom which is called superexchange mechanism.

28

5. Results and discussions

5.1 Computational detailsThe 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 theplane waves was set to be 500eV. Perdew-Bruke-Enzerhof’s (PBE) versionof the generalized gradient approximation (GGA)[24] was used to describethe exchange correlation density functional. For better descriptions of theelectronic 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 ofelectronic steps calculations was 10−7 eV.

It’s reasonable to set the G-type (Rock-Salt) antiferromagnetic (AFM) or-der as the initial magnetic configuration in all calculations. The experimentsshowed BiFeO3 has a approximate G-type AFM order modified with long-wavelength spin spiral structure[8] and total residual magnetic moments causedby weak ferromagnetism[46][12].

In geometry relaxations,a 5×5×5 Monkhorst-Pack k-grid mesh [47] cen-tered at Γ was used to sample the Brillouin zone. The Fermi smearing methodwas employed for the total energy calculations with the width of 0.05 eV. Theatomic positions were fully relaxed using the conjugated gradient method forthe energy minimization with a criterion that requires the force on each atomsmaller than 0.001 eV/Å.

A well known defect of DFT calculation is that it always underestimatethe band gap of system with localized d orbitals, which can be remedied byintroducing on-site Coulomb repulsion between d states in DFT+U[31] ap-proach. In this work, the GGA+U method was implemented in The simplifiedapproach was introduced by Dudarev et al[32].In this approach only the effec-tive Hubbard parameter Ue f f =U−J is meaningful, thus the J parameter wasset 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 ofk-points on the string was set to be 8. The center of the cell in direct latticecoordinates which the total dipole-moment of the cell is calculated is specifiedin a way that all ions keep in the same side. The ionic contribution to polariza-tion 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 polar-ization and the ionic polarization. The R3C phase (centrosymmetric phase) ofBiFeO3 is a candidate for paraelectric phase in high temperature. We used theR3C phase with the same cell volume as R3C phase as the midpoint betweenR3C phase and its enantiomorphic counterpart -R3C phase. The transitionpath is generated by linear interpolation from R3C phase to R3C phase in ourcalculations.

To investigate the non-collinear magnetism of BiFeO3, magnetic anisotropyenergy of BiFeO3 was calculated in two approaches: non-self-consistent andself-consistent approach. The spin-orbital coupling was switched on in all cal-culation. In the self-consistent approach, the residual total magnetic momentand canted alignment of magnetic moments were obtained. The MAE calcula-tions were done with k-points mesh in different sizes to study the convergenceof MAE. The electronic step convergence threshold of 10−7eV was accurateenough for the small value of MAE. The spin-spiral dispersion calculationswere performed in the first Brillouin zone along [111] and [10-1] direction ofthe rhombohedral lattice.

5.2 Structure of BiFeO3The ground state of BiFeO3 in room temperature has a rhombohedral symme-try and R3c space group. The R3c primitive unit cell is derived from the cubicperovskite by rotating the two neighbouring FeO6 octahedra about the [111]axis in the opposite ways and displacing ions along [111] axis. The ideal per-ovskite lattice has a Pm3m space group. The rotation of FeO6 alone reducesthe symmetry from Pm3m to centrosymmetric R3c space group. While thedisplacement along [111] reduces the symmetry from Pm3m to rhombohedralR3m. The combination of these two deformations lowers the symmetry to R3Cand makes BiFeO3 polarized and highly distorted. In Table.5.1 the importantstructural parameters of BiFeO3 R3c phase obtained from geometry relaxationare shown for Ue f f = 0, 2, 4 eV. Comparing with calculated parameters fromNeaton et al’s [12] work and Ravindran et al’s[16] work and measured datafrom Kubel et al’s[48] work, the effective U value of 2 eV is found to be mostappropriate for geometry optimization. The volume of primitive cell is under-estimated by the LSDA in the work of Neaton et al [12] and is overestimatedby GGA in this work and the work of Ravindran et al[16].The value of Uaffects parameters of BiFeO3 to a very limited extent. With U varying from0eV to 2eV, the lattice constant a is increased only by 0.24%. Noticeably thestructural parameters of R3c BiFeO3 do not increase monotonically with Uvalue, which means there is no linear relation between the structure and theon-site Coulomb repulsion. In the geometry relaxation of R3c phase, the celland atom position are relaxed together.

30

Structure a(Å) α() V0(Å3) 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 rhombohedralprimitive cell a is given in Å. The α is the rhombohedral angle, a rhombohedralprimitive cell with α=60 is equivalent to an ideal cubic perovskite.

In Table5.1, the structure of R3c phase from this study is very close to theresult from the work of Neato et al[12]. The rhombohedral angle α of R3cphase 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 comparingwith the ideal perovskite. Whereas the primitive cell of R3c is compressedalong the [111] axis with α larger than 60.

The schematic representation of primitive cell of R3c phase is shown inFig5.1. The two FeO6 octahedra were rotated along [111] in opposite direc-tions which makes the [111] a C3 rotation axis. From the left panel of Fig.5.1we 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 R3c BiFeO3 is less distorted than R3c phase, the center ofO6 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 Oatoms has two values: 2.93 Å and 2.79 Å even without ions displacement.This small distortion on O6 octahedron possibly originates from the chemicalenvironment’s asymmetry caused by the rotation of FeO6 about [111].

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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 R3c BiFeO3 from GGA+U

(Ue f f = 2eV ) calculation.

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

5.3 Electronic properties of BiFeO3The electronic band structure of ferroelectric R3c BiFeO3 is presented in theFig.5.3. The R3c BiFeO3 is insulating with a gap of 1.03 eV in the GGAcalculation. The band gap is increased to 1.83 eV the Ue f f value of 2 eV andincreased to 2.22 eV by Ue f f =4 eV. The R3c BiFeO3 has an direct band gapin GGA calculation at the Z point. The band gap becomes indirect in GGA+Ucalculations, the bottom of conducting band locates between the F point andΓ point, the top of valence band locates on the Z point.

The total density of states (TDOS) from GGA and GGA+U calculations forR3c BiFeO3 are shown in Fig.5.4. The spin-up channel and spin-down channelare symmetric to each other for TDOS of BiFeO3 and Fe atoms which is a signof the antiferromagnetic order of the R3c BiFeO3. We can see that the bandgap 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 phaseof BiFeO3 in GGA+U calculation with Ue f f = 2 eV is shown in the left panelof Fig.5.5. The valence bands of R3c in the range from -6 eV to Fermi levelare 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 ofFeO6 octahedron. The band with lower energy locating in the range from -10

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Figure 5.4. Total density of states of BiFeO3 and Fe, Bi and O atoms in R3c BiFeO3for Ue f f = 0, 2, 4 eV.

eV to -8 eV is the 6s band of Bi atoms. Bi 6s band has no hybridization whichis consistent with the existence of Bi lone pair[49] in BiFeO3. In the atomicorbital of Bi, the 6p orbitals has higher energy than the 6s orbitals. But the 6pbands 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 andthen becomes Bi3+ ion with only 6s electrons in its valence band in the R3cBiFeO3. Further, the acquiring of electrons can account the abundance of 2pstates of O atoms in the valence band.

The magnetic moments of Fe atoms from GGA and GGA+U calculationsare in good agreement with the experimental value of 3.75 µB measured fromlow-temperature neutron-diffraction measurements[8]. The local magneticmoments of Fe atom is ± 3.728, ± 3.980 and ± 4.138 µB respectively forUe f f = 0, 2 and 4 eV. Since the hybridization between Fe 3d band and O 2pband, the non-magnetic O atoms become magnetic with average magnetic mo-ments 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 withthe Fe atom it hybridizes with. From magnetic moments from calculationswith different Ue f f values, it is found that the introducing of on-sine Coulombrepulsion weakens the hybridization between Fe and O and then increase themagnetic moment of Fe and decrease the magnetic moment of O.

The PDOS of R3c BiFe3 is shown in the right panel of Fig.5.5. Similar tothe R3c phase, the top area of valence band of R3c is also mainly composedof the 2p band of O atoms and the 3d band of Fe atoms, and the 6s band ofBi atoms does not hybridize with other band. But, the magnetic configurationof R3c phase is ferromagnetic which is totally different with the R3c phase,the Fe atoms have magnetic moments of 1.52 µB . The ferromagnetism ofR3c causes the asymmetry of the valence band in the PDOS of Fe atoms as

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Figure 5.5. The projected density of states of Bi, Fe and O atoms of R3c and R3cBiFeO3.

shown in Fig.5.5. The hybridization between O 2p band and Fe 3d band makesO atoms’ magnetic moments in the same direction as Fe atoms and causesasymmetry of the valence band in the PDOS of O atoms. The 6s band of Biatoms keeps symmetric which confirms again there is no hybridization existsbetween Bi 6s band and other bands. Notably, the R3c BiFeO3 is not insultingin our calculation. Considering the enlargement effect of U parameters onthe band gap of R3c BiFeO3, the conductivity of R3c phase may be causedby the most common defect of DFT calculation: it always underestimate theband gap. Thus, the calculation with Ue f f = 4 eV is done to investigate thisproblem. The TDOS of the R3c with Ue f f = 4 eV is shown in Fig.5.6. WithUe f f = 4 eV, the R3c BiFeO3 is insulting with a narrow band gap of 1 eV andthe magnetic moments of Fe atoms increase to 4.27 µB. Thus, the U value of 4eV is more appropriate to describe the electronic properties of the paraelectricR3c BiFeO3.

5.4 Electric polarizationThe Berry phase theory[36][50][51] is used to calculate the polarization ofR3c BiFeO3. Due to the polarization lattice mentioned in Chapter 3.1, thepolarization of R3c BiFeO3 may have several possible values spaced by a fixedpolarization quantum. Since the R3c is centrosymmetric and paraelectric, itshould has a zero polarization in the absence of external electric field. Thus,the real polarization value of R3c phase is the polarization difference betweenthe R3c phase and the R3c phase.

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Figure 5.6. Total density of states for the R3c 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 R3c are symmetric about the plane perpendicularto [111], whereas the symmetry is broken in the −R3c and R3c.

In our calculation, the transition path from R3c to R3c was generated bylinear interpolation and 3 intermediate structures between R3c and R3c werechosen to performed the berry phase calculation. To keep the consistency inour calculation, the primitive cell of R3c has the same cell size and shape asthe R3c in our berry phase calculation. Not like the R3c in Section 5.3 , thisR3c is insulting with a narrow gap of 1 eV as shown in Fig. 5.7, which isnecessary to perform berry phase calculation. This R3c BiFeO3 has the AFMorder which indicates that the origin of AFM order is not the ions displacementalong [111].

The results are shown in Fig. 5.9. The polarization of the R3c BiFe3 is103.5 µC/cm2 and the polarization quantum is about 123 µC/cm2 calculation.Our result of polarization is consistent with experimental polarization value of100 µC/cm2 [15] for [111] direction. For the polarization quantum, we seea discrepancy between result from this study and calculation by Neaton etal [12]. One possible reason for the discrepancy is that the centrosymmetricstructure in this study is the R3C phase whereas the cubic phase is used inothers’ calculation. The choice of different transition path is another possiblereason.

Our calculation confirms the ferroelectric order in R3c BiFe3 and predicts apolarization value close to measured value. To understand the ferroelectricitybetter and calculate the polarization more accurate, a more reasonable switch-ing path is very necessary.

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Figure 5.7. Total density of states of the R3c BiFeO3 with the same cell size and shapeas the R3c BiFeO3

Figure 5.8. The lateral view of the primitive cell of the -R3c, R3c and R3c BiFeO3along the [111] direction

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

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

5.5 Magnetic anisotropyThe non-collinear magnetic property is reflected in the magnetic anisotropyenergy. The spin spiral order is suppressed in our calculation, which is con-sistent with the case in epitaxial thin films. Firstly, the convergence of MAEin collinear spins and magnetic moments in non-collonear spins with differentk 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 be-tween two configurations: magnetic moments parallel to [111] and magneticmoments parallel to an axis perpendicular to [111]. As shown in Fig. 5.10, a5×5×5 k points mesh with 125 points is enough for both MAE and magneticmoments to converge.

The single-ion anisotropy is studied by calculate the dependence of energyof 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 spinsare opposite to each other and tend to be perpendicular to [111] axis as shownin Fig.5.12(a) . In the (111) plane, the energy does not change with the orien-tation of collinear spins. The MAE and total magnetic moment of noncollinearspin configuration were calculated self-consistently. As shown in Fig.5.11(b),the noncollinear spins have the same magnetic easy plane the (111) planeas the collinear case, the total magnetic moment also has the maximum inthe easy plane. The MAE of non-collinear spins originate from both single-ion anisotropy and anisotropic superexchange, thus the MAE of non-collinearspins is larger than collinear spins. The single-ion anisotropy’s contribution toMAE is 124 µV /unit cell and the contribution of superexchange is 89µV /unitcell. In the non-collinear configuration, M1 and M2 are canted away from thecollinear direction as shown in Fig. 5.12, the total residual magnetic momentis 47 mµB/unit cell and lies in the magnetic easy (111) planes. This residualtotal 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.

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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 antiferromagneticvector L = M1−M2, and the sum of two magnetic moments is defined asthe ferromagnetic vector M = M1 +M2. Then the DM Hamiltonian can bewritten as

HDM =−2D · (L×M) (5.2)

The HDM will reduce the total energy with particular D and then make the twomagnetic 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 relationIn this section, we firstly mainly discuss the difference of antiferromagneticphase and ferromagnetic phase of BiFeO3. In our calculations, the AFM andFM phase have the same crystal structure in order to exclude any unexpecteddisturbance.

The phonon dispersion of the two phases along [111], [110], [110] and[112] are presented in Fig. 5.13. The AFM BiFeO3 obtained from calculationagrees well with the result from inelastic x-ray scattering[52]. The phononband structure of FM BiFeO3 has imaginary frequency near the gamma pointalong the [110], [110] and [112] direction, which indicates the instability ofits structure.

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Figure 5.12. (a) The magnetic moments of the two Fe atoms M1 and M2 are oppositeto each other and perpendicular to [111] in collinear configuration of spins. (b) In thenon-collinear configuration, magnetic moments of Fe atoms are canted. M1 and M2do not cancel each other completely, the residual total magnetic moment Mr lies in aplane 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

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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 thatis equalivent to [101] in rhombohedral lattice. The energy of FM BiFeO3 de-creases as the wave vector q increase along the [101] direction and equals tothe 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 AFMand FM phase.

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6. Summary and Outlook

In this thesis, the structure, electronic and magnetic properties of BiFeO3 werestudied in detials by the density functional theory (DFT).

In Chapter 2, 3, 4, the theoretical background related to this study are in-troduced. Investigating the physical properties of materials is a many bodyproblem that cannot be solved analytically. The DFT simplify the many bodyproblem 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 theexchange-correlation functional Exc[n] is not given by DFT itself. The localspin density approximation (LSDA) and generalized gradient approximation(GGA) are commonly used from of the exchange-correlation functional. Toimprove the accuracy, meta-GGA and hybrid functional were introduced inDFT calculation. The on-site Coulomb repulsion was included in the LDA+Umethod, in which the strong correlation between localized orbitals are de-scribed more reasonable. The wave functions are expanded by plane wavebasis and the potential in core region is replaced by a pseudopotential to makethe DFT calculation practical on computer. The phonon dispersion is derivedby solving the dynamic matrix D(k). The modern polarization theory explainsthe multiple values of electric polarization in infinite system like periodic lat-tice. The electronic contribution to the polarization is calculated by treating itas a Berry phase of valence wave functions. Heisenberg model of magnetismexplains the formation of different magnetic orders. Non-collinear magneticphenomenon 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 ef-fect into the calculations. With this effective U value, the calculated structuralparameters for BiFeO3 are in good agreement with experiments and the bandstructure of both R3c and R3c are insulting. The Berry phase calculations pre-dicted a polarization of 103.5 µC/cm2 for R3c BiFeO3 which agrees well withexperimental measured value for single crystal sample. The (111) planes werefound to be magnetization easy plane for R3c BiFeO3 for collinear spins. Inthe self-consistent calculation, the two spins of neighbor Fe atoms were cantedaway from antiparallel arrangement, the residual total magnetic moment of 47mµB/unitcell indicates weak ferromagnetism in R3c BiFeO3. The phononcalculations point out the instability of R3c BiFeO3 with ferromagnetic order.In the spin spiral dispersion, it was found that the energy of FM BiFeO3 de-creases as the spin wave vector q increases from gamma point to the boundaryof the first Brillouin zone along [101] direction, whereas the energy of AFMBiFeO3 increases and has the same value as FM at q=(0.34 0 -0.34).

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

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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 fortheir kind assistance and encouragement. I can’t make so much progress inacademics and personal abilities in this two years without them. Specially, Iwant to thank to my wife Xiaoyu Wen for her loving considerations and greatconfidence in me all through these two years when we were separated in twodifferent countries.

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References

[1] P. Hohenberg and W. Kohn. Inhomogeneous electron gas. Phys. Rev.,136:B864–B871, Nov 1964.

[2] Smolenskiı. 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 inmultiferroic 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. Originof magnetic anisotropy and spiral spin order in multiferroic bifeo3. AppliedPhysics 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 bifeo3multiferroic 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. Journalof Applied Physics, 97(9):093903, 2005.

[14] YP Wang, GL Yuan, XY Chen, JM Liu, and ZG Liu. Electrical and magneticproperties of single-phased and highly resistive ferroelectromagnet bifeo3ceramic. Journal of Physics D: Applied Physics, 39(10):2019, 2006.

[15] Delphine Lebeugle, Dorothée Colson, A Forget, and Michel Viret. Very largespontaneous electric polarization in bi fe o 3 single crystals at room temperatureand 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. Theoreticalinvestigation 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. InMathematical 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 electrongas. 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 gradientapproximation made simple. Physical review letters, 77(18):3865, 1996.

[25] John P Perdew and Yue Wang. Accurate and simple analytic representation ofthe electron-gas correlation energy. Physical Review B, 45(23):13244, 1992.

[26] Fabien Tran and Peter Blaha. Accurate band gaps of semiconductors andinsulators with a semilocal exchange-correlation potential. Physical reviewletters, 102(22):226401, 2009.

[27] Axel D Becke. Density-functional thermochemistry. iii. the role of exactexchange. The Journal of chemical physics, 98(7):5648–5652, 1993.

[28] Axel D Becke. Density-functional exchange-energy approximation with correctasymptotic behavior. Physical review A, 38(6):3098, 1988.

[29] Seymour H Vosko, Leslie Wilk, and Marwan Nusair. Accurate spin-dependentelectron liquid correlation energies for local spin density calculations: a criticalanalysis. Canadian Journal of physics, 58(8):1200–1211, 1980.

[30] Chengteh Lee, Weitao Yang, and Robert G Parr. Development of thecolle-salvetti correlation-energy formula into a functional of the electrondensity. Physical review B, 37(2):785, 1988.

[31] Vladimir I Anisimov, Ferdi Aryasetiawan, and AI Lichtenstein. First-principlescalculations of the electronic structure and spectra of strongly correlatedsystems: 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: Anlsda+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 andmagnetoelectric coupling in bismuth ferrite. Phys. Rev. B, 71:060401, Feb 2005.

[40] Matthias Menzel, Yuriy Mokrousov, Robert Wieser, Jessica E. Bickel, ElenaVedmedenko, Stefan Blügel, Stefan Heinze, Kirsten von Bergmann, AndréKubetzka, and Roland Wiesendanger. Information transfer by vector spinchirality 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 projectoraugmented-wave method. Physical Review B, 59(3):1758, 1999.

[44] Georg Kresse and Jürgen Furthmüller. Efficient iterative schemes for ab initiototal-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 liquidmetals. Physical Review B, 47(1):558, 1993.

[46] GP Vorob’ev, AK Zvezdin, AM Kadomtseva, Yu F Popov, VA Murashov, andDN Rakov. Possible coexistence of weak ferromagnetism and spatiallymodulated 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 ferroelasticmonodomain crystal of the perovskite bifeo3. Acta Crystallographica SectionB: Structural Science, 46(6):698–702, 1990.

[49] Ram Seshadri and Nicola A Hill. Visualizing the role of bi 6s âlone pairsâ in theoff-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 quantityand its relation to surface charge. Physical Review B, 48(7):4442, 1993.

[51] Raffaele Resta. Macroscopic polarization in crystalline dielectrics: thegeometric 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 toepitiaxial-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 ofantiferromagnetics. Journal of Physics and Chemistry of Solids, 4(4):241 – 255,1958.

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