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Claude EdererFirst principles studies of multiferroic materials
First principles studies of multiferroic materials
Claude EdererSchool of Physics, Trinity College Dublin
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...
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!
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!
Magneto-electric multiferroics
Claude EdererFirst principles studies of multiferroic materials
Magneto-electric multiferroics = ferromagnetic + ferroelectric
●Ferromagnetic:
M
●Ferroelectric:
P
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!
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● ...
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
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
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
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
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...
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
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)
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?
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:
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
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.
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.
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.
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
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)
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
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...
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, ... ???
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
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
+ + +- --
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
+ + +- --
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
+ + +- --
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
+ + +- --
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
+ + +- --
+ + +- --
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:
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
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
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))
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
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):
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
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
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))
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°
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°
?
?
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)
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
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
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)
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)
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:
Magnetic anisotropy in BiFeO3
Claude EdererFirst principles studies of multiferroic materials
~ 2meV (LSDA)P In-plane 6-fold degeneracy
(bulk):
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):
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
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°
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)
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)
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
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
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...
Magnetically induced ferroelectricity
Claude EdererFirst principles studies of multiferroic materials
● Two examples
a) Spiral multiferroics: TbMnO3
b) Ferroelectricity from collinear magnetic order: HoMnO3
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):
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)
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
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
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
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)
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
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
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
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
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
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
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
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
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
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
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