First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First...
Transcript of First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First...
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Claude EdererFirst principles studies of multiferroic materials
First principles studies of multiferroic materials
Claude EdererSchool of Physics, Trinity College Dublin
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Overview
Claude EdererFirst principles studies of multiferroic materials
1) Introduction to multiferroic materials
● Why first principles calculations?
2) Density functional theory
3) Examples:
a) BiFeO3
● Electric polarization
● Strain dependence
● Coupling between polarization and magnetism?
● Computational design of new multiferroic materials
b) Other examples...
![Page 3: First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First principles studies of multiferroic materials What is a multiferroic? Hans Schmid: “A](https://reader036.fdocuments.us/reader036/viewer/2022062402/5f0863437e708231d421c2b6/html5/thumbnails/3.jpg)
Introduction and definitions
Claude EdererFirst principles studies of multiferroic materials
What is a multiferroic?Hans Schmid: “A material that combines two (or more) of the primary ferroic order parameters in one phase”
In practice often: multiferroic = (anti-)ferromagnetic + ferroelectric = magnetic ferroelectric
Important: ● switchable domains (change in point
symmetry)● not necessarily coupled!
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Introduction and definitions
Claude EdererFirst principles studies of multiferroic materials
What is a multiferroic?Hans Schmid: “A material that combines two (or more) of the primary ferroic order parameters in one phase”
In practice often: multiferroic = (anti-)ferromagnetic + ferroelectric = magnetic ferroelectric
Related but different: magneto-electric effect(electric field induces magnetization, magnetic field induces electric polarization)
Important: ● switchable domains (change in point
symmetry)● not necessarily coupled!
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Magneto-electric multiferroics
Claude EdererFirst principles studies of multiferroic materials
Magneto-electric multiferroics = ferromagnetic + ferroelectric
●Ferromagnetic:
M
●Ferroelectric:
P
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Magneto-electric multiferroics
Claude EdererFirst principles studies of multiferroic materials
Magneto-electric multiferroics = ferromagnetic + ferroelectric
●Ferromagnetic:
M
●Ferroelectric:
P
●Domains:
●Hysteresis:
Non-volatile data-storage!
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Magneto-electric multiferroics
Claude EdererFirst principles studies of multiferroic materials
• Coexistence of ferroelectric, ferroelastic and magnetic order
→ Interesting cross-correlations between polarization, magnetization, and strain!
From: Spaldin/Fiebig: “The renaissance of magneto-electric multiferroics”, Science 15, 5733 (2005)
Possible Applications:● magneto-electric RAM (electric
write/magnetic read)● four-state memory● ...
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Some history
Claude EdererFirst principles studies of multiferroic materials
Known magnetic ferroelectrics:1961: Smolenskii et al.: mixed perovskites (e.g. Pb(Fe
2/3W
1/3)O
3, Pb(Fe
1/2Nb
1/2)O
3)
1963: Smolenskii/Kiselev: BiFeO3
1963: Bertaut et al.: hexagonal RMnO3 (e.g.
YMnO3, HoMnO
3)
1966: Ascher/Schmid: Boracites M3B
7O
13X
(e.g. Ni3B
7O
13I)
1968: Eibschuetz/Guggenheim et al.: BaMF4
(e.g. BaMnF4 BaNiF
4)
History of magnetoelectric (ME) effect1894: First conjecture about ME effect by Pierre Curie
1956: Landau/Lifshitz formulate symmetry requirements for ME effect (concept of time reversal symmetry)
1959: Dzyaloshinskii predicts ME effect in Cr
2O
3
1960: Experimental confirmation by Astrov (ME)
E
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
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Recent boom
Claude EdererFirst principles studies of multiferroic materials
→ Large polarization and (small) magnetization above room temperature
→ Small Polarization created by non-centrosymmetric magnetic order
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Classification of magnetic ferroelectrics
Claude EdererFirst principles studies of multiferroic materials
One multiferroic is not necessarily equal to another multiferroic !
YMnO3
TbMn2O5
BiFeO3
)1 Ferroelectricity independent of magnetism● Boracites: Ni3B7O13I, Ni3B7O13Cl, Co3B7O13I, ...● “Doped” multiferroics: Pb(Fe2/3W1/3)O3, Pb(Fe1/2Nb1/2)O3, ...● “Lone pair” ferroelectrics: BiFeO3, BiMnO3, ...● “Geometric” ferroelectrics
● proper: BaMF4 (M=Mn, Fe, Co, Ni)● improper: YMnO3, HoMnO3, ... (hexagonal manganites)
)2 Ferroelectricity induced by ...● ...magnetic order: TbMnO3, TbMn2O5, Ni3V2O8, CuFeO2, CoCr2O4,... ● ...charge order”: LuFe2O4, Pr1-xCaxMnO3 (?)
BaNiF4
CoCr2O4
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Why first principles calculations?
Claude EdererFirst principles studies of multiferroic materials
● 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
First principles: start directly from fundamental laws of Physics, without model assumptions or fitting parameters
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Overview
Claude EdererFirst principles studies of multiferroic materials
1) Introduction to multiferroic materials
● Why first principles calculations?
2) Density functional theory
3) Examples:
a) BiFeO3
● Electric polarization
● Strain dependence
● Coupling between polarization and magnetism?
● Computational design of new multiferroic materials
b) Other examples...
![Page 13: First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First principles studies of multiferroic materials What is a multiferroic? Hans Schmid: “A](https://reader036.fdocuments.us/reader036/viewer/2022062402/5f0863437e708231d421c2b6/html5/thumbnails/13.jpg)
Density functional theory
Claude EdererFirst principles studies of multiferroic materials
Interacting many-body problem: Effective single particle problem:
mapping
exact for ground state!
“Exchange-correlation potential”(has to be approximated)
Hohenberg/Kohn 1964, Kohn/Sham 1965, Nobel Prize in Chemistry 1998 for Walter Kohn
● Facilitates quantitative predictions of materials properties● Provides powerful analysis-tool for electronic structure
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The Hohenberg-Kohn Theorems
Claude EdererFirst principles studies of multiferroic materials
The problem:
Effort to calculate increases exponentially with N→ only possible for small molecules (N ~10)
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The Hohenberg-Kohn Theorems
Claude EdererFirst principles studies of multiferroic materials
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?
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The Kohn-Sham equations
Claude EdererFirst principles studies of multiferroic materials
Idea (Kohn/Sham 1965): construct density from auxiliary non-interacting system with the same ground state density
Interacting system:
Non-interacting system:
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The Kohn-Sham equations
Claude EdererFirst principles studies of multiferroic materials
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
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The local density approximation (LDA)
Claude EdererFirst principles studies of multiferroic materials
Exchange-correlation energy density of a homogeneous electron gas of density n
Expected to be good for not slowly varying densities.
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The local density approximation (LDA)
Claude EdererFirst principles studies of multiferroic materials
Exchange-correlation energy density of a homogeneous electron gas of density n
Extremely successful!
Expected to be good for not slowly varying densities.
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The local density approximation (LDA)
Claude EdererFirst principles studies of multiferroic materials
Exchange-correlation energy density of a homogeneous electron gas of density n
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, ...
Expected to be good for not slowly varying densities.
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Beyond LDA: correlated electrons
Claude EdererFirst principles studies of multiferroic materials
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
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Beyond LDA: correlated electrons
Claude EdererFirst principles studies of multiferroic materials
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 Edc
(shifts relative to “uncorrelated” bands)
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Quantities that can be calculated
Claude EdererFirst principles studies of multiferroic materials
● Charge density, total energies● → energy differences between different structures, forces,
phonons● Spin density for magnetic systems, energy differences between different
magnetic configurations, magnetic anisotropy energies● Single particle band-structure, electronic density of states, (zeroth
approximation for electronic excitation spectra)● Electric polarization, dielectric constants
In addition:● Results can be analyzed in terms of fundamental quantities● “Computer experiments”, with the possibility to control the position of each
individual atom, switch off certain interactions, ...
● Quantitative predictions of materials properties● Powerful analysis-tool for electronic structure
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Overview
Claude EdererFirst principles studies of multiferroic materials
1) Introduction to multiferroic materials
● Why first principles calculations?
2) Density functional theory
3) Examples:
a) BiFeO3
● Electric polarization
● Strain dependence
● Coupling between polarization and magnetism?
● Computational design of new multiferroic materials
b) Other examples...
![Page 25: First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First principles studies of multiferroic materials What is a multiferroic? Hans Schmid: “A](https://reader036.fdocuments.us/reader036/viewer/2022062402/5f0863437e708231d421c2b6/html5/thumbnails/25.jpg)
BiFeO3: A room temperature multiferroic
Claude EdererFirst principles studies of multiferroic materials
● ferroelectric below TE ≈ 1100 K● antiferromagnetic below TM ≈ 600 K● Controversial results about the “spontaneous polarization”:
1970: P = 6 µC/cm2 (single crystals) Teague et al., Solid State Comm. 8, 1073
2003: P = 60 µC/cm2 (thin films) Wang et al., Science 299, 1719
Large P: Effect of strain, defects, impurity phases, ... ???
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Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
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Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
+ + +- --
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Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
+ + +- --
![Page 29: First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First principles studies of multiferroic materials What is a multiferroic? Hans Schmid: “A](https://reader036.fdocuments.us/reader036/viewer/2022062402/5f0863437e708231d421c2b6/html5/thumbnails/29.jpg)
Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
+ + +- --
![Page 30: First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First principles studies of multiferroic materials What is a multiferroic? Hans Schmid: “A](https://reader036.fdocuments.us/reader036/viewer/2022062402/5f0863437e708231d421c2b6/html5/thumbnails/30.jpg)
Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
+ + +- --
![Page 31: First principles studies of multiferroic materialszhiwu/research/slides/B12_mf.pdf · First principles studies of multiferroic materials What is a multiferroic? Hans Schmid: “A](https://reader036.fdocuments.us/reader036/viewer/2022062402/5f0863437e708231d421c2b6/html5/thumbnails/31.jpg)
Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
+ + +- --
+ + +- --
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Electric polarization
Claude EdererFirst principles studies of multiferroic materials
● Finite system:Not applicable within periodic boundary conditions (depends on unit cell choice).
King-Smith/Vanderbilt 1993, Resta 1994: “Modern theory of electric polarization”● Polarization of a bulk solid is a multivalued quantity● Only differences in polarization are meaningful quantities
+ + +- --
+ + +- -- Spontaneous polarization:
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BiFeO3: Electric polarization
Claude EdererFirst principles studies of multiferroic materials
Neaton, Ederer, Waghmare, Spaldin, Rabe, PRB 71, 014113 (2005)
King-Smith/Vanderbilt 1993, Resta 1994
Polarization in bulk periodic solid:
???Problem: undistorted structure metallic in LDA→ need LDA+U
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BiFeO3: Electric polarization
Claude EdererFirst principles studies of multiferroic materials
Neaton, Ederer, Waghmare, Spaldin, Rabe, PRB 71, 014113 (2005)
King-Smith/Vanderbilt 1993, Resta 1994
Polarization in bulk periodic solid:
Problem: undistorted structure metallic in LDA→ need LDA+U
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BiFeO3: Electric polarization
Claude EdererFirst principles studies of multiferroic materials
Neaton, Ederer, Waghmare, Spaldin, Rabe, PRB 71, 014113 (2005)
King-Smith/Vanderbilt 1993, Resta 1994
Polarization in bulk periodic solid:
Problem: undistorted structure metallic in LDA→ need LDA+U→ evaluate polarization for intermediate distortion
PS = 95 µC/cm2
Large intrinsic polarization Ps(bulk) ≈ 95 µC/cm2 ( ≈ Ps(film))
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BiFeO3: Electric polarization
Claude EdererFirst principles studies of multiferroic materials
Neaton, Ederer, Waghmare, Spaldin, Rabe, PRB 71, 014113 (2005)
Born effective charges:
α formal
Bi +6.32 +3
Fe +4.55 +3
O -3.62 -2
Z*Bi ion drives the ferro-
electric distortion
See also: Seshadri/Hill: Visualizing the role of Bi 6s “lone pairs” in the off-center distortion in ferromagnetic BiMnO3, Chem. Mater. 13, 2892 (2001)
Compare with BaTiO3: (Ghosez/Michenaud/Gonze, PRB 58, 6224 (1998))
Ba: Z = 2.75 , Ti: Z = 7.16, O: Z = -5.69/-2.11→ Ti drives the distortion
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Strain effects in thin film ferroelectrics
Claude EdererFirst principles studies of multiferroic materials
Epitaxial thin film growth:
In-plane lattice constant determined by substrate:
→ epitaxial strain
→ can have drastic effects on ferroelectric properties
“Enhancement of ferroelectricity in strained BaTiO3 thin films”, K. J. Choi et al., Science 306, 1005 (2004):
“Room-temperature ferroelectricity in strained SrTiO3”, J. H. Haeni et al., Nature 430, 758 (2004):
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BiFeO3: Effect of epitaxial strain
Claude EdererFirst principles studies of multiferroic materials
Strain dependence:
Theory predictions:● Large intrinsic bulk polarization● Very weak epitaxial strain dependence
[1] Neaton et al. (2002)[2] Bungaro/Rabe (2004)
Symbols: direct calculation, lines: using ceff
c33, c31: piezoelectric constants
n: Poisson ratioε: epitaxial strain
Ederer/Spaldin PRB 71, 224103 (2005)Ederer/Spaldin PRL 95, 257601 (2005)
...but only weak effect in BiFeO3
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BiFeO3: More recent experiments
Claude EdererFirst principles studies of multiferroic materials
● Lebeugle et al., Appl. Phys. Lett. 91, 022907 (2007) : “Very large spontaneous electric polariztion in BiFeO3 single crystals at room temperature and its evolution under cycling fields”
● Kim et al., Appl. Phys. Lett. 92, 012911 (2008) : “Effect of epitaxial strain on ferroelectric polarization in multiferroic BiFeO3 films”
Consistent with results of first principles calculations
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BiFeO3: Magnetic properties
Claude EdererFirst principles studies of multiferroic materials
Bulk: G-type AFM + cycloidal rotation (λ=640nm)
+
Thin films:
From: Lebeugle et al., PRL 100, 227602 (2008)
Wang et al., Science 299, 1719 (2003)
Small magnetization but no cycloidal rotation (Bea et al., Phil Mag. 87, 165 (2007))
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Weak ferromagnetism in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
M ≈ 0.1 µB/Fe
Dzyaloshinskii-Moriya interaction (Moriya 1960):Calculations show:
Antiferromagnetic sub-lattices are canted by ≈ 1°
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Weak ferromagnetism in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
M ≈ 0.1 µB/Fe
Dzyaloshinskii-Moriya interaction (Moriya 1960):
How is the canting coupled to the structural distortions?
D P
Electric-field-induced magnetization switching?
Calculations show:
Antiferromagnetic sub-lattices are canted by ≈ 1°
?
?
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Magneto-structural coupling in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
BiFeO3: two different structural modes!
1. Counter-rotations of oxygen octahedra around [111]
2. Polar displacements along [111]
Both symmetry analysis and first principles calculations show:
DM interactions is generated by oxygen octahedra rotations !
Ederer/Spaldin, PRB 71, 060401 (2005)
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Magneto-structural coupling in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
BiFeO3: two different structural modes!
1. Counter-rotations of oxygen octahedra around [111]
2. Polar displacements along [111]
Both symmetry analysis and first principles calculations show:
DM interactions is generated by oxygen octahedra rotations !
Ederer/Spaldin, PRB 71, 060401 (2005)
Symmetry analysis: L in BFO does not break space inversion symmetry !
L has to change sign under both time and space inversion
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Effect of octahedral rotations
Claude EdererFirst principles studies of multiferroic materials
● ionic displacements corresponding to octahedral rotations:
● DM = 0 if midpoint between magnetic sites is inversion center
● octahedral rotations lift inversion center between B sites→ weak magnetism is induced
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Effect of octahedral rotations
Claude EdererFirst principles studies of multiferroic materials
● ionic displacements corresponding to octahedral rotations:
● DM = 0 if midpoint between magnetic sites is inversion center
● octahedral rotations lift inversion center between B sites→ weak magnetism is induced
Solution:● put magnetic cation on A-site,
(e.g. FeTiO3)→ L is odd under space inversionC. J. Fennie, PRL 100, 167203 (2008)
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Ferroelectric/magnetic domains in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
Polarization along {111} direction → 8 different FE domains
Piezoelectric force microscopy (PFM):
Zavaliche et al., APL 87, 182912 (2005)
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Ferroelectric/magnetic domains in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
Polarization along {111} direction → 8 different FE domains
Piezoelectric force microscopy (PFM):
Zavaliche et al., APL 87, 182912 (2005)
Correlation with magnetic domains?
XLD (PEEM) PFM
10×8 μm2
FE domains:AFM domains (?):
X-ray linear dichroism (XLD) depends on orientation of antiferromagnetic axis:
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Magnetic anisotropy in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
~ 2meV (LSDA)P In-plane 6-fold degeneracy
(bulk):
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71° switching
Magnetic anisotropy in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
109° switching
~ 2meV (LSDA)P
Magnetic moments want to be perpendicular to P→ changing the direction of P will affect magnetic order
In-plane 6-fold degeneracy (bulk):
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Electric-field switching of AFM domains
Claude EdererFirst principles studies of multiferroic materials
Zhao et al., Nature Materials 5, 823 (2006)
71° switching → AFM axis preserved
109° switching → AFM axis changed
● (001)-oriented films have small monoclinic distortion
● 6-fold degeneracy is broken● Calculation: monoclinic strain favors [110]
direction
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Electric-field switching of AFM domains
Claude EdererFirst principles studies of multiferroic materials
Zhao et al., Nature Materials 5, 823 (2006)
71° switching → AFM axis preserved
109° switching → AFM axis changed
● (001)-oriented films have small monoclinic distortion
● 6-fold degeneracy is broken● Calculation: monoclinic strain favors [110]
direction
→ in agreement with exp. observations
1, 2: 109°; 3: 71°; 4: 180°
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Why is it interesting?
Claude EdererFirst principles studies of multiferroic materials
Exchange bias coupling to a ferromagnet:
→ effective electric-field switching of magnetization
Magnetoelectric RA MBibes/Barthelemy, Nature Materials 7, 425 (2008)
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Very recent work
Claude EdererFirst principles studies of multiferroic materials
Exchange bias demonstrated recently for BiFeO3/CoFeB heterostructures:Bea et al., PRL 100, 017204 (2008)
“Electric field control of local ferromagnetism using a magnetoelectric multiferroic”, Chu et al., Nature Materials 7, 478 (2008)
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Computational design of novel multiferroics
Claude EdererFirst principles studies of multiferroic materials
Layered double perovskite structure:
Predicted ground state properties:
• Ps ≈ 80 µC/cm2
• M = 2µB/formula unit
Baettig/Spaldin, APL 86, 012505 (2005); Baettig/Ederer/Spaldin, PRB 72, 257601 (2005)
Bi 2FeCrO 6: A ferrimagnetic ferroelectric
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Computational design of novel multiferroics
Claude EdererFirst principles studies of multiferroic materials
Layered double perovskite structure:
Predicted ground state properties:
• Ps ≈ 80 µC/cm2
• M = 2µB/formula unit
Baettig/Spaldin, APL 86, 012505 (2005); Baettig/Ederer/Spaldin, PRB 72, 257601 (2005)
Bi 2FeCrO 6: A ferrimagnetic ferroelectric
Systematic LSDA+U study for BiFeO3 – Bi2FeCrO6 – BiCrO3 to estimate TC
Mean-field approximation for TC
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Overview
Claude EdererFirst principles studies of multiferroic materials
1) Introduction to multiferroic materials
● Why first principles calculations?
2) Density functional theory
3) Examples:
a) BiFeO3
● Electric polarization
● Strain dependence
● Coupling between polarization and magnetism?
● Computational design of new multiferroic materials
b) Other examples...
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Magnetically induced ferroelectricity
Claude EdererFirst principles studies of multiferroic materials
● Two examples
a) Spiral multiferroics: TbMnO3
b) Ferroelectricity from collinear magnetic order: HoMnO3
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Orthorhombic manganites
Claude EdererFirst principles studies of multiferroic materials
T. Kimura et al.: PRB 68, 060403(R), 2003:RMnO3 (R=La, Pr, Nd, ... , Ho)
Orthorhombically distorted perovskite structure (Pnma symmetry):
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Orthorhombic manganites
Claude EdererFirst principles studies of multiferroic materials
T. Kimura et al.: PRB 68, 060403(R), 2003:RMnO3 (R=La, Pr, Nd, ... , Ho)
Orthorhombically distorted perovskite structure (Pnma symmetry):
Example 1:TbMnO
3 – representative
for “spiral multiferroics” (non-collinear)
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Orthorhombic manganites
Claude EdererFirst principles studies of multiferroic materials
T. Kimura et al.: PRB 68, 060403(R), 2003:RMnO3 (R=La, Pr, Nd, ... , Ho)
Orthorhombically distorted perovskite structure (Pnma symmetry):
Example 1:TbMnO
3 – representative
for “spiral multiferroics” (non-collinear)
Example 2:HoMnO
3
– collinear magnetic order breaks inversion symmetry
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TbMnO3: Experiment
Claude EdererFirst principles studies of multiferroic materials
● Small Polarization below TC ~ 28K
● Polarization can be rotated from c to a by magnetic field
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Ferroelectricity induced by spiral magnetic ordering
Claude EdererFirst principles studies of multiferroic materials
Example - frustrated Heisenberg spin chain:
Mostovoy, PRL 96, 067601 (2006)Free energy (Lifshitz invariant):
Periodicity depends on relative strength of various coupling constants→ often incommensurate
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Microscopic mechanism
Claude EdererFirst principles studies of multiferroic materials
● Spin-current modelKatsura/Nagaosa/Balatsky, PRL 95, 057205 (2005)“electronically driven”
● Inverse DM interactionSergienko/Dagotto, PRB 73, 094434 (2006)“lattice driven”
In both cases:
Spin-orbit coupling → P typically μC/m2 (BaTiO3: 25 μC/cm2)
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First principles calculations
Claude EdererFirst principles studies of multiferroic materials
Malashevich/Vanderbilt, PRL 101, 037210 (2008):
● Simplified commensurate spin order k=1/3 (exp. k=0.28)
● Highly accurate calculations including spin-orbit coupling (SOC)
Results:● Without SOC: P = 0● With SOC, no ionic relaxation: P = 32 μC/cm2
● SOC + ionic relaxations: P = -467 μC/cm2
● Exp.: P = -600 μC/cm2
Polarization mainly “lattice-driven”, but not fully compatible with simple DM model
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Alternative mechanism without SOC
Claude EdererFirst principles studies of multiferroic materials
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
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Alternative mechanism without SOC
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
Claude EdererFirst principles studies of multiferroic materials
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Alternative mechanism without SOC
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
Claude EdererFirst principles studies of multiferroic materials
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Alternative mechanism without SOC
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
Claude EdererFirst principles studies of multiferroic materials
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Alternative mechanism without SOC
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
Claude EdererFirst principles studies of multiferroic materials
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Alternative mechanism without SOC
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
Claude EdererFirst principles studies of multiferroic materials
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Alternative mechanism without SOC
Sergienko/Sen/Dagotto, PRL 97, 227204 (2006)
E-type AFM in orthorhombic manganites (e.g. HoMnO3)
Relevant free energy invariant:
Double exchange model (virtual hopping):
➔ FM bonds: p >
0 (less distorted)
➔ AFM bonds: ap
< 0 (more distorted)
~S
eg
E
t2g
Mn3+: d4
P
Interplay of hopping, octahedral rotations and E-type AFM leads
to electric polarization
Claude EdererFirst principles studies of multiferroic materials
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HoMnO3: First principles calculations
Picozzi et al., PRL 99, 227201 (2007)
Sizable polarization ~6μC/cm2
(not spin-orbit related!)
Not confirmed by experiment, yet, but difficult to prepare single domain state.
Claude EdererFirst principles studies of multiferroic materials
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HoMnO3: First principles calculations
Picozzi et al., PRL 99, 227201 (2007)
Sizable polarization ~6μC/cm2
(not spin-orbit related!)
Not confirmed by experiment, yet, but difficult to prepare single domain state.
Similar mechanism might be at work in RMn
2O
5
(R=Tb, Ho, Y, ..)
Claude EdererFirst principles studies of multiferroic materials
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Summary
Claude EdererFirst principles studies of multiferroic materials
● First principles calculations allow to make quantitative predictions of materials properties and provide a powerful analysis tool
● Examples:✔ Polarization in bulk BiFeO
3 is large and only slightly affected by
epitaxial strain ✔ Weak magnetization in thin films is coupled to antiferrodistortive
counter-rotations of oxygen octahedra✔ Electric field induced switching of AFM domains can be explained by
change in magneto-crystalline anisotropy✔ New “designer multiferroics” can be predicted✔ Polarization in TbMnO
3 mostly lattice-driven
✔ “Exchange-striction” can cause significant polarization even for collinear magnetic order