First principles studies of multiferroic materials Claude Ederer - PowerPoint PPT Presentation

First principles studies of multiferroic materials Claude Ederer School of Physics, Trinity College Dublin edererc@tcd.ie Claude Ederer First principles studies of multiferroic materials

Overview 1) Introduction to multiferroic materials ● Why first principles calculations? 2) Density functional theory 3) Examples: a) BiFeO 3 ● Electric polarization ● Strain dependence ● Coupling between polarization and magnetism? ● Computational design of new multiferroic materials b) Other examples... Claude Ederer First principles studies of multiferroic materials

Introduction and definitions What is a multiferroic? Hans Schmid: “A material that combines two (or more) of the primary ferroic order parameters in one phase” Important: ● switchable domains (change in point symmetry) ● not necessarily coupled! In practice often: multiferroic = (anti-)ferromagnetic + ferroelectric = magnetic ferroelectric Claude Ederer First principles studies of multiferroic materials

Introduction and definitions What is a multiferroic? Hans Schmid: “A material that combines two (or more) of the primary ferroic order parameters in one phase” Important: ● switchable domains (change in point symmetry) ● not necessarily coupled! In practice often: multiferroic = (anti-)ferromagnetic + ferroelectric = magnetic ferroelectric Related but different: magneto-electric effect (electric field induces magnetization, magnetic field induces electric polarization) Claude Ederer First principles studies of multiferroic materials

Magneto-electric multiferroics Magneto-electric multiferroics = ferromagnetic + ferroelectric ● Ferromagnetic: M ● Ferroelectric: P Claude Ederer First principles studies of multiferroic materials

Magneto-electric multiferroics Magneto-electric multiferroics = ferromagnetic + ferroelectric ● Ferromagnetic: ● Domains: M ● Hysteresis: ● Ferroelectric: P Non-volatile data-storage! Claude Ederer First principles studies of multiferroic materials

Magneto-electric multiferroics • Coexistence of ferroelectric, ferroelastic and magnetic order From: Spaldin/Fiebig: “The renaissance of magneto- electric multiferroics”, Science 15, 5733 (2005) Possible Applications: → Interesting cross-correlations between ● magneto-electric RAM (electric polarization, magnetization, and strain! write/magnetic read) ● four-state memory ● ... Claude Ederer First principles studies of multiferroic materials

Some history History of magnetoelectric (ME) effect Known magnetic ferroelectrics: 1894 : First conjecture about ME effect by 1961 : Smolenskii et al.: mixed perovskites Pierre Curie (e.g. Pb(Fe 2/3 W 1/3 )O 3 , Pb(Fe 1/2 Nb 1/2 )O 3 ) 1956 : Landau/Lifshitz formulate symmetry 1963 : Smolenskii/Kiselev: BiFeO 3 requirements for ME effect (concept of time 1963 : Bertaut et al.: hexagonal R MnO 3 (e.g. reversal symmetry) YMnO 3 , HoMnO 3 ) 1959 : Dzyaloshinskii predicts ME effect in 1966 : Ascher/Schmid: Boracites M 3 B 7 O 13 X Cr 2 O 3 (e.g. Ni 3 B 7 O 13 I) 1960 : Experimental confirmation by Astrov 1968 : Eibschuetz/Guggenheim et al.: Ba M F 4 (ME) E (e.g. BaMnF 4 BaNiF 4 ) 1961 : Reciprocal (ME) H effect measured by Rado et al. But: small effects, mostly low temperatures, scarcity of materials, lack of microscopic understanding Recently: improved theoretical understanding, thin film preparation, new experimental techniques Claude Ederer First principles studies of multiferroic materials

Recent boom → Large polarization and → Small Polarization created (small) magnetization by non-centrosymmetric above room temperature magnetic order Claude Ederer First principles studies of multiferroic materials

Classification of magnetic ferroelectrics 1 Ferroelectricity independent of magnetism ) BiFeO 3 ● Boracites: Ni 3 B 7 O 13 I, Ni 3 B 7 O 13 Cl, Co 3 B 7 O 13 I, ... ● “Doped” multiferroics: Pb(Fe 2/3 W 1/3 )O 3 , Pb(Fe 1/2 Nb 1/2 )O 3 , ... ● “Lone pair” ferroelectrics: BiFeO 3 , BiMnO 3 , ... ● “Geometric” ferroelectrics ● proper: Ba M F 4 ( M =Mn, Fe, Co, Ni) ● improper: YMnO 3 , HoMnO 3 , ... (hexagonal manganites) YMnO 3 2 Ferroelectricity induced by ... ) ● ...magnetic order: TbMnO 3 , TbMn 2 O 5 , Ni 3 V 2 O 8 , CuFeO 2 , CoCr 2 O 4 ,... ● ...charge order”: LuFe 2 O 4 , Pr 1-x Ca x MnO 3 (?) BaNiF 4 TbMn 2 O 5 CoCr 2 O 4 One multiferroic is not necessarily equal to another multiferroic ! Claude Ederer First principles studies of multiferroic materials

Why first principles calculations? First principles : start directly from fundamental laws of Physics, without model assumptions or fitting parameters ● Diverse materials science requires a theoretical approach that is able to resolve differences between different materials ● Provide reference values for experimental data (make predictions) ● Rationalize experimental observations Claude Ederer First principles studies of multiferroic materials

Overview 1) Introduction to multiferroic materials ● Why first principles calculations? 2) Density functional theory 3) Examples: a) BiFeO 3 ● Electric polarization ● Strain dependence ● Coupling between polarization and magnetism? ● Computational design of new multiferroic materials b) Other examples... Claude Ederer First principles studies of multiferroic materials

Density functional theory Interacting many-body problem: Effective single particle problem: mapping exact for ground state! “Exchange-correlation potential” (has to be approximated) ● Facilitates quantitative predictions of materials properties ● Provides powerful analysis-tool for electronic structure Hohenberg/Kohn 1964, Kohn/Sham 1965, Nobel Prize in Chemistry 1998 for Walter Kohn Claude Ederer First principles studies of multiferroic materials

The Hohenberg-Kohn Theorems The problem: Effort to calculate increases exponentially with N → only possible for small molecules (N ~10) Claude Ederer First principles studies of multiferroic materials

The Hohenberg-Kohn Theorems The problem: Effort to calculate increases exponentially with N → only possible for small molecules (N ~10) Hohenberg/Kohn 1964: ● All ground state properties of an interacting many-electron system are uniquely determined by the electron density ● The correct ground state density minimizes the total energy functional Density replaces many-body wavefunction as central quantity of interest But how to obtain the density? Claude Ederer First principles studies of multiferroic materials

The Kohn-Sham equations Idea (Kohn/Sham 1965): construct density from auxiliary non-interacting system with the same ground state density Interacting system: Non-interacting system: Claude Ederer First principles studies of multiferroic materials

The Kohn-Sham equations Idea (Kohn/Sham 1965): construct density from auxiliary non-interacting system with the same ground state density Interacting system: Non-interacting system: Iterate until self-consistency Still missing: expression for Claude Ederer First principles studies of multiferroic materials

The local density approximation (LDA) Exchange-correlation energy density of a homogeneous electron gas of density n Expected to be good for not slowly varying densities. Claude Ederer First principles studies of multiferroic materials

The local density approximation (LDA) Exchange-correlation energy density of a homogeneous electron gas of density n Expected to be good for not slowly varying densities. Extremely successful! Claude Ederer First principles studies of multiferroic materials

The local density approximation (LDA) Exchange-correlation energy density of a homogeneous electron gas of density n Expected to be good for not slowly varying densities. Extremely successful! Problems: ● Underestimates band gaps in many semiconductors ● Not adequate for strongly correlated d or f electrons (eventually predicts metallic instead of insulating ground states) → Improved xc-functionals: Generalized Gradient Approximation (GGA), Exact exchange, hybrid functionals, GW, ... Claude Ederer First principles studies of multiferroic materials

Beyond LDA: correlated electrons Hubbard model : ● Competition between hopping (kinetic energy) and electron-electron interaction ● Contains main physics that dominates properties of many d and f electron systems ● But: extremely simplified, empirical parameters Claude Ederer First principles studies of multiferroic materials

Beyond LDA: correlated electrons Hubbard model : ● Competition between hopping (kinetic energy) and electron-electron interaction ● Contains main physics that dominates properties of many d and f electron systems ● But: extremely simplified, empirical parameters → Combine Hubbard-type interaction with LDA/DFT: LDA+ U (Anisimov et al. 1991) ● Leads to correct insulating ground state for many transition metal oxides ● Important: U dependence (basis set dependent parameter), double counting term E dc (shifts relative to “uncorrelated” bands) Claude Ederer First principles studies of multiferroic materials