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Magnetic interactions from first-principles · 2016-06-08 · Types of magnetic interactions The...
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Magnetic interactions from first-principles
Mikhail Katsnelson
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Outline
1. Introduction 2. Time-dependent DFT and magnetic
susceptibility 3. Exchange interactions from first principles 4. Beyond DFT: correlated systems and
LDA+DMFT 5. Applications: Fe, Ni, Gd, NiO, CrO2… 6. Dzyaloshinskii-Moriya interactions 7. Application: Molecular magnets 8. Orbital and spin contributions 9. Towards consequent theory of ultrafast spin
dynamics
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Epigraphs
To the theoretical physicists, ferromagnetism presents a number of very interesting, unsolved and beautiful challenges. Our challenge is to understand why it exists at all.
Make things as simple as possible but not simpler
(Feynman Lectures on Physics)
(A. Einstein)
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Magnetite – first known magnet
Very complicated structure, still a lot of open questions
Two types of Fe sites (tetra and octa); Metal-insulator transition; Charge ordering; Role of orbital degrees of freedom; Half-metallicity...
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Types of magnetic ordering
Sometimes very complicated
α-Mn
Textbook wisdom
Spin spirals
γ-Fe UO2
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Relation to superconductivity and other phenomena
Simplified phase diagram of Cu-O high-Tc superconductors
Layered cobaltates NaxCoO2
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Types of magnetic interactions
The first term: exchange interactions (Heisenberg model) Quantum, nonrelativistic (Coulomb interaction plus Pauli principle). Determine the type of magnetic ordering (mostly) The second term: magnetic anisotropy Quantum, relativistic (due to spin-orbit interaction). At least, second-order in SOC. Determine “practical” magnetism (hard and soft magnetic materials, hysteresis loop, etc.) The third term: Dzyaloshinskii-Moriya interactions Quantum, relativistic (due to spin-orbit interaction). First-order in SOC but require broken inversion symmetry. Responcible for weak FM, skyrmiones etc.
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General formulation
System of interacting electrons (many-body problem) + crystal potential
External strong time-dependent laser field
(nonequilibrium problem)
Temperature effects (thermal bath, open system, basic statistical mechanics)
Collect all difficulties of modern theoretical
physics
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Levels of description
- Macroscopic (LLG equations + temperature balance, etc.) - Microscopic, classical Heisenberg model - Microscopic, quantum itinerant-electron model -Ab initio, time-dependent density functional
Multiscale problem
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Time-dependent DFT
SE for many-body wave function in configurational space is replaced by single-particle nonlinear self-consistent equation
Spinor
B is self-consistent magnetic field
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Simplifications
Adiabatic approx.: Vxc and Bxc are the same as in the equilibrium + local (spin) density approx.
n,m are charge and spin densities
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Linear response: magnetic susceptibility
MIK & Lichtenstein, JPCM 16, 7439 (2004)
At the same time (Runge-Gross theorem, 1984) in TD-DFT
A response of effective system of noninteracting Kohn-Sham particles (Liu & Vosko 1989 for magnetic case)
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Linear response: magnetic susceptibility II
Rigorous exprression
Adiabatic approximation plus LSDA:
Transverse susceptibility is separated from (longitudinal spin + charge) susceptibilities
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Transverse susceptibility
Local Stoner parameter
Kohn-Sham states
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Longitudinal susceptibility
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Separation of magnon poles
After rigorous manipulations
Magnon pole
Im part corresponds to Stoner damping
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Alternative definition of exchanges
Static susceptibility
The first way (poles of susceptibility) corresponds Liechtenstein, MIK & Gubanov, J. Phys. F 1984, the second way (static suscept.) Bruno, PRL 2003. The expressions for stiffness constant coincide and are rigorous within the adiabatic approximation + LSDA
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Nonlocal corrections to magnon stiffness
MIK & Antropov, PRB 67, 140406 (2003) Exchange and correlation in spiral state of
homogeneous electron gas Angular gradient
corrections
Corrections to stiffness constant
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Stiffness constants for Fe and Ni
Fe: LSDA 239
with gradient corrections 251 experiment 280 - 310
Ni: LSDA 692 with gradient corrections 735 experiment 550-630
(in meV/Å2)
Corrections are quite small
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Stoner damping in Fe and Ni
Antropov, Harmon, Smirnov, JMMM 200, 148 (1999)
fcc-Ni
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Magnetic force theorem
(Lichtenstein & MIK 1984)
Total energy in DF
Variation
at fixed potential due to change of potential
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Magnetic force theorem II
- Torque can be written in terms of variation of the density of states - Decomposition of the torque in pair terms gives exchange integrals (LK) - These exchange parameters are local (near given magnetic configuration) - Adding constrain to stabilize rotated configuration gives exchange parameters (Bruno) Exchange parameters for d metals are strongly non- Heisenbergian (depend on magnetic configuration) (Turzhevskii, Lichtenstein & MIK, Fiz. Tverd. Tela 1990)
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Example: magnetism of Fe, Co, Ni
Iron Cobalt Nickel
Ferromagnetism of iron is known from ancient times
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Itinerant-electron ferromagnetism at finite temperatures
Stoner
T=0
T<Tc
T>Tc
Heisenberg Spin-fluctuation
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Stoner criterion
Equation for the Curie temperature:
f(E) Fermi function
Stoner parameter ≈ 0.9 eV for all 3d metals; DOS is crucially important
If Fe would be Stoner magnet it would have TC ≈ 4000 K (in reality 1043 K)
In reality, Tc is determined by spin fluctuations, That is, exchange parameters
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Iron: some details
Crystal field splitting DOS for nonmagnetic bcc Fe
Stoner criterion is fulfilled due to eg states only; they should play a special role in magnetism of Fe (Irkhin, Katsnelson, Trefilov,
JPCM 5, 8763 (1993))
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Non-Heisenberg character of exchange interactions in Fe and Ni
Rotation of a central spin: magnetic moment is not constant, energy change is not cosine
Electronic structure is angle-dependent
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Iron: detailed analysis
t2g are itinerant electrons providing (Heisenberg-like) RKKY exchange with Friedel oscillations; eg are more correlated providing (non-Heisenberg) “double exchange” typical for narrom-band systems
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Problem with DFT: coexistence of localized and itinerant behavior
4f electrons are normally pure localized but not 3d
Local magnetic moments do exist above TC (Curie-Weiss law, spectroscopy, neutrons…) d electrons are itinerant (FS, chemical bonding, transport…)
Iron, majority spin FS
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From atomic state to itinerant
Experiment: disappearance of multiplets
Calculations: increase of hybridization
Blue line: exact diagonalization for free atom
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Dynamical Mean Field Theory I
A natural generalization of the familiar MFT to the problem of electrons in a lattice
A.Georges, G.Kotliar, W.Krauth and M.Rozenberg, Rev. Mod. Phys. ‘96
Key idea: take one site out of a lattice and embed it in a self-consistent bath = mapping to an effective impurity problem
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Dynamical Mean Field Theory II
( )ττ ′−0G
( ) ( )∑Ω=
BZ
knn ikGiG
ωω ,ˆ1ˆ
( ) ( ) ( )nnn iiGi ωωω Σ+= −− ˆˆˆ 110G
Σ Σ Σ
Σ Σ
Σ Σ Σ
Σ U QMC ED
DMRG IPT FLEX
( ) ( ) ( )nnnnew iGii ωωω 110
ˆˆˆ −− −=Σ G
Single Impurity Solver
W. Metzner and D. Vollhardt (1987) A. Georges and G. Kotliar (1992)
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Ferromagnetism of transition metals: LDA+DMFT
-8 -6 -4 -2 0 20.0
0.5
1.0
1.5
2.0
2.5
LDA
DMFT
PES
Ni: LDA+DMFT (T=0.9 Tc)
EF
Den
sity
of s
tate
s, e
V-1
Energy, eV
0 2 4 60.0
0.5
1.0
1.5
τ, eV-1
<S(
τ)S(
0)>
Ferromagnetic Ni DMFT vs. LSDA: • 30% band narrowing • 50% spin-splitting reduction • -6 eV sattellite
LDA+DMFT with ME J. Braun et al PRL (2006)
Lichtenstein, MIK, Kotliar, PRL (2001)
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Orbital magnetic moments
For FexCo1-x alloys
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LDA+Disordered Local Moments
J. Staunton and B. Gyorffy PRL69, 371 (1992)
The best first-principle Spin-fluctuation model with classical moments
DLM
EXP
DLM
EXP
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DMFT Effective Magnetic Moments: T>Tc W exp eff loc DLM Tc exp
Fe 3.13 3.09 2.8 1.96 1900 1043Ni 1.62 1.5 1.3 1.21 700 631
0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4 1,6 1,8 2,0 2,20,0
0,2
0,4
0,6
0,8
1,0
1,2
0,0
0,2
0,4
0,6
0,8
1,0
1,2
χ(T)M(T)
M(T) and χ(T): LDA+DMFT
Ni
Fe
χ-1M
eff2 /3
T c
M(T
)/M(0
)
T/Tc
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ARPES for iron
Agreement is not bad (much better than LDA/GGA) but essentially worse than in nickel. Correlations in iron are not quite local
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ARPES for 3d metals
Black – spin up, red – spin down Upper panel – exper,lower - DMFT
Variation of U does not help
too much for Fe
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Why Ni is more local than Fe?
S. Hershfield http://www.phys.ufl.edu/fermisurface
Nickel is almost half-metallic: majority-spin FS almost coincides with the boundaries of the Brillouin band
But the difference for minority spin is even more dramatic
Occupations for majority (minority) electrons 5 means full occupation
Fe: 4.6 (2.34) Ni: 4.82 (4.15)
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Why Ni is more local than Fe II Friedel oscillations originating from FS are much weaker in nickel
As a result: Magnons are much softer in Fe than in Ni (Curie temp. Higher but magnon frequencies lower)
Fe Co Ni
The softer magnons the stronger nonlocal e-m intercation
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Exchange and Functionals
Magnetic force theorem
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LDA+DMFT (Lichtenstein & MIK 1997, 1998,1999; Anisimov et al 1997)
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Exchange interactions from DMFT Heisenberg exchange:
Magnetic torque:
Exchange interactions:
Spin wave spectrum:
MIK & Lichtenstein Phys. Rev. B 61, 8906 (2000)
Non-collinear magnetism :
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Alternative view First- and second- order smallness in theta angle!
Total energy corrections by diagram technique neglecting vertex corrections → our exchanges
Exact within DMFT (local self-energy!)
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Applications
For Fe (and Ni)
quite small difference between DFT and DMFT
Nontrivial: electronic structure is very
different! Error cancellation?!
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Applications II
Does not follow a naive formula Difference between Mott and charge transfer insulator
Gd: also, DFT works quite good
NiO: not too big difference between DMFT and LDA + U
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Applications III
Half-metallic FM DMFT shows
non-quasiparticle states in the gap
MIK et al, RMP 80, 315 (2008)
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Applications IV
Important consequences from DMFT contributions to exchange
Without magnetic polarization of oxygen FM state is unstable within DMFT (but not in simpler
approaches)
Direct exchange also plays an important role
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Dzialoshinskii-Moriya interactions
MIK, Kvashnin, Mazurenko & Lichtenstein, PRB 82, 100403 (2010)
LDA+U
DM interactions (weak FM, etc.)
Small rotations
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Dzialoshinskii-Moriya interactions II
Starting from collinear configuration
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Applications to La2CuO4
Canting angle 0.005 Exper. 0.003
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FeBO3
A novel exper. technique to measure DM vector and not only canting angle (resonant X-ray scattering)
Agrees well with exper.
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Molecular magnets
Example: V15 AFM ground state S = 1/2
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LDA+U calculations
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LDA+U calculations II
Exact diagonalization for Heisenberg model
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Mn12: full calculations
Motivation The prototype molecular magnet
Dimension of Hilbert space:
(2ˣ2+1)8(2ˣ3/2+1)4=108
A real challenge!
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Mn12: full calculations II
Inelastic netron scattering data: cannot be explained without strong DM interactions (MIK, Dobrovistki & Harmon, PRB 1999) Eight-spin model: S = ½ dimers from S=2 and S=3/2 Dimensionality of Hilbert space decreases to 104 Cannot be justified quantitatively!
Full LDA+U calculations plus Lanczos ED
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Mn12: full calculations III
Plus anisotropy tensors...
No fitting parameters at all – not so bad!
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Mn12: full calculations IV
Also, thermodynamic quantities can be calculated
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Spin and orbital contributions
Rotation operator involves both spin and orbital rotations
No smallness of SOC is assumed
Calculate the change of energy at small rotations, map to the classical spin orbital Hamiltonian
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Spin and orbital contributions II
Decomposition of exchange parameters and similar for other
interactions Hopping can be excluded using the Dyson equations
E.g., for DM interactions
and similar for exchanges. Important for actinides, RE, and 3d systems with unquenched orbital moments (e.g., CoO)
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Ultrafast magnetism: Examples
Nickel Koopmans et al PRL 2005 Orthoferrites Kimel et al Nature 2005
Gadolinium Melnikov et al PRL 2003
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Ultrafast magnetism: a theory
Consider dynamics of Baym-Kadanoff-Keldysh countour
Path integral over Grassmann variables
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Ultrafast magnetism: a theory II
Introduce rotations
Expand effective actions up to the second order in “Holstein-Primakoff” fields ξ, ξ*
Integrate over Grassman variables neglecting vertex corrections
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Ultrafast magnetism: a theory III
General expression of nonlocal in time exchange interactions in terms of Beym-Kadanoff-Keldysh Green’s functions. E.g., time-dependent stiffness constant:
Additional terms (twist exchange) of the structure
(at equilibrium forbidden by time-reversal symmetry)
The first step is done, a lot of things to do
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Beyond the talk Finite-temperature effects
Ab initio spin dynamics for real systems Intermediate level: TB spin dynamics And many, many specific applications to real materials
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Collaboration
Recent: A. Lichtenstein and S. Brener (Hamburg) A. Secchi and A. Rudenko (Nijmegen) V. Mazurenko (Ekaterinburg) Ya. Kvashnin and O. Eriksson (Uppsala) and many other people involved in development of the formalism and calculations for specific materials in 1987-2013, esp. V. Antropov (Ames) and D. Boukhvalov (Seoul)
Thank you for your attention