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Transport properties calculated by means of the Kubo formalism
Hubert Ebert, Diemo Ködderitzsch, Sergiy Mankovskyy, Kristina Chadova, and Sebastian Wimmer
Ludwig-Maximilians-Universität München, Department Chemie, Physikalische Chemie,Butenandtstrasse 5-13, 81377 München, Germany
Outline
● Introduction
● Kubo-Středa vs. Kubo-Bastin
● Kubo vs. Boltzmann formalism
● Symmetry predicted properties
● Inclusion of temperature
● Summary and outlook
2
AHE
2
2
anomalous Ettingshausen
effect
anomalous Ettingshausen
effect
Righi-Leduc effect (thermal Hall)
Righi-Leduc effect (thermal Hall)
SHE & EE
AHE
ANE
ANE
3SHE & EE
2
ISHE & IEE
3ISHE & IEE
3SNE
SNE
3
ISNE
ISNE
4
?"spin spin Hall"?
?"spin spin Hall"?
FM
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NPSMP2015, Workshop, Hubert Ebert 22015-06-17
Charge, heat and spin transportwith emphasis on spin-orbit induced effects
• Charge
• Heat current density
• Spin
• Electric field
• Temperature gradient
• Fictitious field coupling to spin
Study of pure systems and disordered alloys
FerromagnetFerromagnet
Paramagnet Paramagnet
Longitudinal effects: AMR & ASE
AHE
SNE
ANE
SHE
Goal: Investigation treating all microscopic contributions on equal footing on first-principles level
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 3
Hierarchy of transport equations
Kubo
Bastin
Kubo-Středa
Kubo-Greenwood
Independent electron approximation, ω = 0
T = 0K
Retaining symmetric part only
see also: Ködderitzsch, Lowitzer, Staunton, Ebert, phys. stat. sol. (b) 248, 2248 (2011).
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 4
SPR-KKR-CPA electronic structure calculations
SPR: Dirac equation within LSDA
KKR: Green function via multiple scattering theory
CPA: Coherent potential approximation for disorder
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NPSMP2015, Workshop, Hubert Ebert 52015-06-17
Boltzmann-formalism applied to AgxPd1-x
Residual resistivity (T=0K)
W. H. Butler et al., PRB 29, 4217 (1984)
Neglecting scattering-in term
Bloch spectral function of Cu0.80Pd0.20
group velocity
life time
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NPSMP2015, Workshop, Hubert Ebert 62015-06-17
Kubo-Greenwood equation and Vertex corrections
relativistic quantum numbers
Implementation within KKR-CPA
Butler, PRB 31, 3260 (1985) (non-relativistic)Banhart et al., SSC 77, 107 (1991) (fully-relativistic)Turek et al., PRB 65, 125101 (2002) (LMTO-CPA)
See also: Velicky, PR 184, 614 (1969)
Vertex corrections (VC)
account for scattering-in processes
vertex correction
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 7
Kubo-Středa vs. Kubo-Bastin
approach
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NPSMP2015, Workshop, Hubert Ebert 82015-06-17
Expt.: Matveev et al., Fiz. Met. Metalloved 53, 34 (1982)Theo.: Lowitzer et al., PRL 105, 266604 (2010)
Anomalous Hall conductivityin ferromagnetic alloys
KKR-CPA results based on Kubo-Středa equation
Pd Fe NiPd
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NPSMP2015, Workshop, Hubert Ebert 92015-06-17
SHE for 4d- and 5d-transition metal alloys
KKR-CPA results based on Kubo-Středa equation
Guo et al., PRL 100, 096401 (2008)Guo, JAP 105, 07C701 (2009)Yao et al., PRL 95, 156601 (2005)
intrinsic SHE of pure elements
Pt Au Au Ag
}Lowitzer et al., PRL 106, 056601 (2011)
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NPSMP2015, Workshop, Hubert Ebert 102015-06-17
Transverse charge transport CoxPd1-x alloys
● No direct relation between AHC and ANC as functions of x
● AHC shows sign change, while ANC does not
● ANC: Maximum at x ≈ 0.2 in line with behaviour of ρiso
, AMR ratio & Sxx
Anomalous Hall conductivity Anomalous Nernst conductivity
Expt.: Jen et al., JAP 76, 5782 (1994)
S. Wimmer, D. Ködderitzsch, and H. Ebert, Phys. Rev. B 89, 161101(R) (2014)
FerromagnetFerromagnet
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NPSMP2015, Workshop, Hubert Ebert 112015-06-17
Transverse spin transport in AuxCu1-x
(Thermal) Spin Nernst conductivitySpin Hall conductivity
S. Wimmer, D. Ködderitzsch, K. Chadova, and H. Ebert, Phys. Rev. B 88, 201108(R) (2013)
Paramagnet Paramagnet
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NPSMP2015, Workshop, Hubert Ebert 122015-06-17
Kubo-Bastin formalism
settingT = 0K
• numerical difficulties (energy derivative) • integral over δ-function like terms
integration in the complex plane
inclusion of vertex corrections numerical effort
I Turek, J Kudrnovský and V Drchal, PRB 89, 064405 (2014)
Similar approach and implementation within TB-LMTO
EDOS
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NPSMP2015, Workshop, Hubert Ebert 132015-06-17
Kubo-Středa: S.Lowitzer, D.Ködderitzsch, H.Ebert, PRL 105, 266604 (2010)
Anomalous Hall conductivity in ferromagnetic alloys
KKR-CPA results based on Bastin and Kubo-Středa equation ( numerical test for the equivalency )
with vertex corrections(total)
without vertex corrections(coherent part)
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 14
Spin Hall conductivity for transition metal alloys
KKR-CPA results based on Bastin and Kubo-Středa equation ( numerical test for the equivalency )
with vertex corrections(total)
without vertex corrections(coherent part)
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 15
Kubo vs. Boltzmann formalism
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NPSMP2015, Workshop, Hubert Ebert 162015-06-17
Connection with Boltzmann Formalism
Kubo-Greenwood equation within KKR-CPA
Neglecting the vertex corrections givesBoltzmann equation without scattering-in term
Inverse lifetime
Boltzmann equation including scattering-in term
Butler, PRB 31, 3260 (1985)
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NPSMP2015, Workshop, Hubert Ebert 172015-06-17
Connection with Boltzmann Formalism
Kubo-Greenwood equation within KKR-CPA
Neglecting the vertex corrections givesBoltzmann equation without scattering-in term
Inverse lifetime
Boltzmann equation including scattering-in term
Butler, PRB 31, 3260 (1985)
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NPSMP2015, Workshop, Hubert Ebert 182015-06-17
Decomposition of Spin Hall conductivityvia scaling behaviour of individual contributions
Ansatz using scaling behaviour
linear relation on both sides of alloy system for composition
Extrapolation to
KKR-CPA results for Au1-x
Ptx: skewness factor
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NPSMP2015, Workshop, Hubert Ebert 192015-06-17
Description of the (Spin) Hall effect
Kubo-Středa linear response
extrinsicincoherent
intrinsiccoherent
Decomposition
side-jump
skew
scalinglaws
vertexcorrections
scattering-interms
skew
transverse
any system
Berrycurvature
intrinsic
pure systems
longitudinal
Boltzmann
longitudinal
dilute alloys
transverse
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 20
Ferromagnetic host doped with 3d impurities with concentration of 1 at.%
Skew-scattering contribution to the AHE
Anomalous Hall angle (skew-scattering contribution)
Approach Geometry SOC group
Method A Boltzmann Full potential Pauli Jülich (Blügel)
Method B Boltzmann ASA Dirac Halle (I. Mertig)
Method C Kubo ASA Dirac Munich (H. Ebert)
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 21
Paramagnetic host doped with 3d impurities with concentration of 1 at. %
Anomalous Hall angle (skew-scattering contribution)
Spin Hall angle
Approach Geometry SOC group
Method A Boltzmann Full potential Pauli Jülich (Blügel)
Method B Boltzmann ASA Dirac Halle (I. Mertig)
Method C Kubo ASA Dirac Munich (H. Ebert)
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NPSMP2015, Workshop, Hubert Ebert 222015-06-17
Anomalous Hall conductivity in ferromagnetic alloys
Contributions to the incoherent part of the conductivity tensorIncoherent part (extrinsic)
Incoherent part of Fermi sea term is negligible
justification for Kubo-Středa Boltzmann scattering
No vertex corrections for the Fermi sea term !
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 23
Spin Hall conductivity for transition metal alloys
Contributions to the incoherent part of the conductivity tensorIncoherent part (extrinsic)
Incoherent part of Fermi sea term is negligible
justification for Kubo-Středa Boltzmann scattering
No vertex corrections for the Fermi sea term !
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NPSMP2015, Workshop, Hubert Ebert 242015-06-17
AHE – Boltzmann vs Kubo formalism
longitudinal conductivity
Boltzmann-based calculations:Gradhand, Fedorov, Mertig, unpublished (2013)
Comparison of results for varying concentration
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 25
Symmetry predicted properties
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NPSMP2015, Workshop, Hubert Ebert 262015-06-17
Kleiner's scheme to get the shape of the electrical conductivity tensor
for unitary operators u:
for anti-unitary operators a:
same transformation behavior for thermal transport → same tensor shapes
W. H. Kleiner, Phys. Rev. 142, 318 (1966)
Pseudoalgorithm
● determine symmetry of system● loop over symmetry operations
● set up system of linear eqs. inelements
● solution gives restrictions● element is linear combination of
other elements● element is its negative
→ element is zero
Only the magnetic Laue group has to be considered
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NPSMP2015, Workshop, Hubert Ebert 272015-06-17
Electrical conductivity - tensor shapes
class a) contains time reversal T
Results obtained by analytic computation using computer algebra system (CAS)
Electrical conductivity tensor
Thermo-electrical response tensor
Non-magnetic materials Magnetic materials
class c) contains combined operations a=v T
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NPSMP2015, Workshop, Hubert Ebert 282015-06-17
Transverse conductivity and magneto-optics in non-collinear antiferromagnets
Mn3Ir – a prototype non-collinear antiferromagnet
● Cu3Au structure
● moments in (111) plane (Kagome lattice)
● magnetic space group: R3m'
Prediction of anomalous Hall effect (AHE)
and magneto-optical Kerr effect (MOKE)
● based on analysis of electronic structure
Chen, Niu, and MacDonald, PRL 112, 017205 (2014)
● Natural consequence of Kleiner's tablesfor the shape of the conductivity tensor
Kleiner, PR 142, 318 (1966)
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NPSMP2015, Workshop, Hubert Ebert 292015-06-17
Mn3Ir – electrical conductivity
Electrical conductivity tensor
numerical work based on Kubo-Středa equation
● confirms tensor shape
● Anomalous Hall conductivity
275 (Ω cm)-1 this work218 (Ω cm)-1 Chen et al. (2014)
comparable in size to Fe, Co, and Ni
Smrčka and Středa, JPC 10, 2153 (1977)Lowitzer et al., PRL 105, 266604 (2010)
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 30
Mn3Ir – optical conductivity and Kerr angle
● same tensor shape as for bcc-Fe, comparable in magnitude
Mn3Ir
bcc Fe
S. Wimmer, et al., unpublished (2015)
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 31
Mn3Ir – Mn L
2,3 X-ray absorption spectra
● spectra by superposition of site-resolved abs. coeffs.
● incidence : [111] vs. direction of (polar geometry)
● same total absorption
● in both cases XMCD (larger for polar geometry)
● Results questions the XMCD sum rules
11
2
2
33
Wimmer, et al., unpublished (2015)
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NPSMP2015, Workshop, Hubert Ebert 322015-06-17
Extended Kleiner's scheme to get the shape of the spin-polarisation conductivity tensor
[1] V. Bargmann, E. P. Wigner, Proc. Natl. Acad. Sci. U.S.A. 34, 211 (1948)[2] A. Vernes, B.L. Györffy, P. Weinberger, Phys. Rev. B 76, 012408 (2007)[3] S. Lowitzer, Ködderitzsch, H. Ebert, Phys. Rev. B 82, 140402 (2010)
for unitary operators u:
for anti-unitary operators a:
Using a relativistic spin polarization operator [1,2,3]:
Dirac matrix
Only the magnetic Laue group has to be considered
S. Wimmer et al., arXiv:1502.04947, PRB RC accepted (2015)
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NPSMP2015, Workshop, Hubert Ebert 332015-06-17
Electrical and spin-polarized conductivity tensor
e.g.: Au4Sc
e.g.: Pt3Ge
e.g.: Au
e.g.: FM bcc Fe
S. Wimmer et al., arXiv:1502.04947, PRB RC accepted (2015)
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NPSMP2015, Workshop, Hubert Ebert 342015-06-17
Electrical conductivity & DOS of Au1-x
PtxSc
Au2Pt
2Sc
4/m1'
Diagonal conductivity Density of states
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NPSMP2015, Workshop, Hubert Ebert 352015-06-17
Additional transverse elements for 4/m1'
Au4Mn - structure
paramagnetic (Au1-x
Ptx)
4Sc - alloy
4/m1'
xy
z
z x
y
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NPSMP2015, Workshop, Hubert Ebert 362015-06-17
Longitudinal spin polarized currents
paramagnetic ferromagnetic4/m1' 4/mm'm'
Spin Hall effect and longitudinal spin currentin ferromagnet
andparamagnet
caused by spin-orbit interaction
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NPSMP2015, Workshop, Hubert Ebert 372015-06-17
Longitudinal spin polarized currents in 4/m1'
Au4Mn - structure
paramagnetic (Au1-x
Ptx)
4Sc - alloy
4/m1'
xy
z
z x
y
S. Wimmer et al., arXiv:1502.04947, PRB RC accepted (2015)
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NPSMP2015, Workshop, Hubert Ebert 382015-06-17
Longitudinal spin polarized currents in 2/m1'
Au4Mn - structure
paramagnetic (Pt1-x
Aux)
3Ge - alloy
2/m1'
S. Wimmer et al., (unpublished)
Pt3Ge - structure
y
xz
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 39
Inclusion of temperature
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NPSMP2015, Workshop, Hubert Ebert 402015-06-17
Electron-Phonon coupling
Selfenergy
Eliashberg function
Phonon dispersion relation of Cu
Renormalisation of electron structure
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NPSMP2015, Workshop, Hubert Ebert 412015-06-17
Electron-phonon self energy k for fcc-Pb
Minar et al., JESRP 184, 91 (2011)
Experiment: at T = 8K F. Reinert et al., PRL 91, 186406 (2003)
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NPSMP2015, Workshop, Hubert Ebert 422015-06-17
ARPES from Pb(110)
Without SE Including SE Experiment
Calculated ARPES spectra
Expt: F. Reinert et al. PRL 91, 186406 (2003)
Ehν = 21.1 eVMinar et al., JESRP 184, 91 (2011)
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NPSMP2015, Workshop, Hubert Ebert 432015-06-17
Alloy analogy model for thermal vibrations
Representation of thermal vibrations
by temperature dependent, quasi-static, descrete set of dispacements
Multi-component CPA equations CPA
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NPSMP2015, Workshop, Hubert Ebert 442015-06-17
Models for thermal displacements
Fixing the descrete set of dispacements
via temperature dependent root square displacement
root square displacement from Debye model
root square displacement from phonon calculations
CPA
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NPSMP2015, Workshop, Hubert Ebert 452015-06-17
Alloy analogy model for thermal spin fluctuations
Representation of thermal spin fluctuations
by temperature dependent, quasi-static, descrete
set of non-collinear spin orientations
CPA
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 46
Spin fluctuations from Monte Carlo simulations
is defined on the grid:
Fitting of Weiss field parameter
weighting factor at temperature T
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NPSMP2015, Workshop, Hubert Ebert 472015-06-17
Temperature dependent resisitivity of ferromagnetic solids – bcc-Fe
Comparison of different Combination of thermal
models of spin disorder lattice vibrations and
spin fluctuations
● Transverse spin fluctuations important for spin disorder● Impact of thermal lattice vibrations and spin fluctuations
are not additive
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 48
Resistivity: theory vs experiment
Ni fcc: resistivity vs temperature
Fe bcc: resistivity vs temperature
● Fe: MC M(T): magnetic fluctuation effect is overestimated
● Crucial role of M(T) dependence → discrepancies between the resistivity results based on MC and experimental M(T)
● Ni: Longitudinal fluctuations should be taken into account near TC
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 49
Gilbert damping
Gilbert damping for Fe Gilbert damping for Ni
● Fe: comparable contributions of lattice vibrations and spin fluctuations
● Ni: main contribution – from lattice vibrations
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NPSMP2015, Workshop, Hubert Ebert 502015-06-17
Gilbert damping: Temperature effects
Ni-rich Ni1-x
Cux alloys
Expt: Bhagat and Lubitz, Phys. Rev. B 10, 179, (1974)
● Pure Ni: conductivity-like behaviour for low temperatures
● Less than 1 % Cu strongly damp low temperature singularity
● With more than 5 % Cu the temperature dependence is nearly suppressed
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 51
FeRh: electrical resistivity
MC: M(T)/M0 for one Fe sublattice Expt vs theory: resistivity vs T
● Monte Carlo simulations: Temperature dependent magnetization of AFM-aligned sublattices of Fe
● AFM state: Faster decrease of Fe sublattice magnetization (stronger spin fluctuations) → steeper increase of resistivity
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2015-06-17 NPSMP2015, Workshop, Hubert Ebert 52
FeRh: electrical resistivity
FeRh, FM: contributionsFeRh, AFM: contributions
● FM state: Weak contribution to the resistivity from the scatteringdue to lattice vibrations.
● AFM state: stronger spin fluctuations
→ steeper increase of resistivity
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Fe(Rh,Fe): electrical resistivity
Fe(Rh,Fe), FM: contributionsFe(Rh,Fe), AFM: contributions
● FM state: Weak effect of Fe impurities in Rh sublattice
● AFM state: strong effect of Fe impurities in Rh sublattice
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FeRh: Gilbert damping
FeRh, FM
● Main temperature effect: lattice vibrations
Expt: E. Mancini et al. J. Phys. D: Appl. Phys. 46 (2013) 245302
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2H-Fe0.28
TaS2: transport proprties
Electrical resistivity vs T AHE vs T
● Small contribution of spin fluctuations to the electrical resistivity
● AHE: increase of ρxy
(T) at low temperature – due to phonon contribution
● AHE: Crucial effect of temperature induced magnetic disorder for ρxy
(T)
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● Kubo-Středa vs. Kubo-Bastin
● Numerical equivalency demonstrated
● Kubo vs. Boltzmann formalism● Coherent results in the dilute limit
● Symmetry predicted properties● New phenomena identified
● Inclusion of temperature● Description of thermal lattice vibrations
and spin fluctuations via alloy analogy model
Summary
SFB 689
SPP 1538Spin CaloricTransport