1. Institute of Metal Physics
Coulomb correlation effects in electronic structure of iron pnictide superconductors
Vladimir I. Anisimov
Institute of Metal Physics
Ekaterinburg, Russia

2. Outline Dynamical Mean-Field Theory (DMFT)
Combining DMFT with Density Functional Theory methods
(LDA+DMFT)
Realization of the LDA+DMFT computational scheme in
Wannier functions basis
Pnictide superconductors LaOFeAs, BaFe2As2 LiFeAs and LaOFeP investigated within LDA+DMFT method
Summary

3. Dynamical Mean-Field Theory
Object of investigation: interacting lattice fermions
Simplest description – Hubbard model
Unsolvable problem for d≥2
Reason – correlation phenomena
Square lattice, z=4
Approximations need to be made

4. Dynamical Mean-Field Theory
Mapping
Metzner, Vollhardt (1989)
d→∞
Georges, Kotliar (1992)
Jarrell (1992)
mapping onto impurity problem,
self-consistent equations
Effective impurity problem
Real lattice

5. Dynamical Mean-Field Theory
The DMFT mapping means:
Dyson equation for impurity problem:
Dyson equation is used twice in DMFT. First for known self-energy and lattice Green function bath Green function is calculated:
Then after impurity problem solution new approximation for self-energy can be defined:

6. + – LDA+DMFT computational scheme
Start: noninteracting density of states N(ε), initial guess for Σ(iωn)
Self-consistency
check
Impurity solver –
Restricted to single-orbital
or degenerate orbitals case
Can not be applied if orbitals of interest (d-orbitals) strongly hybridize with other electronic states (O2p orbitals) +
Can be applied for Mott insulators with N(ε) – LDA DOS of d-band

7. LDA+DMFT computational scheme
Hilbert transform N(ε)→G(iωn)
can not be applied to
charge transfer insulators
Solution – use full Hamiltonian instead of N(ε)
Effective Hamiltonian
construction
Self-consistency
check
Impurity solver

8. Possible ways to define material-specific :
Effective Hamiltonian construction
Possible ways to define material-specific :
Tight-binding fit to DFT band structure – obtain {tij}
Downfolding tecnique (NMTO)
O. K. Andersen and T. Saha-Dasgupta (2000)
Wannier functions techniques:
(i) Maximally localized generalized Wannier functions
N. Marzari and D. Vanderbilt (1997), F. Lechermann et al (2006)
(ii) Atomic-orbitals projected Wannier functions in
a) LMTO basis set Anisimov et. al. (2005)
b) Pseudopotential basis set
Trimarchi et. al. (2008), Dm. Korotin et. al. (2008)

9. Wannier functions: projection technique
Wei Ku et al. (2002): A good approximation to Maximally localized Wanner functions is projection of trial orbitals onto the subspace of Bloch functions
In our case = site centered pseudoatomic orbitals

10. Wannier functions: applications
I. Kinetic energy term of effective Hamiltonian calculation:
a) Real space
b) Reciprocal space
II. Occupation matrix construction
III. Interaction parameters U and J calculation: a)
b) Constrained DFT, basis - WF

11. LDA+DMFT scheme in Wannier functions basis
LDA calculation – band structure
Orbitals of interest choice (interacting d- or f-orbitals) for projection
Effective Hamiltonian construction for Wannier functions
LDA Effective Hamiltonian
projection
Interaction parameters U and J calculation in constrain DFT
DMFT solution of the problem defined by Hamiltonian

12. dxz-like Wannier function modulus square isosurface:
Wannier functions: NiO example
Wannier states constructed for different energy intervals (Dm.Korotin et al. (2008)): eV eV
dxz-like Wannier function modulus square isosurface:

13. Novel superconductor LaOFeAs
Tc=26K for F content ~11%
Y. Kanamura et al. J. Am. Chem. Soc. 130, 3296 (2008)

14. Novel superconductor LaOFeAs
All bands WF
constrain DFT
U=3.5 eV
J=0.8 eV
Fe3d band
only WF
constrain DFT
U=0.8 eV
J=0.5 eV
d (x2-y2) Wannier functions (WF) calculated
for all bands (O2p,As4p,Fe3d) and
for Fe3d bands only
V.Anisimov et al, J. Phys.: Condens. Matter 21, (2009)

15. Novel superconductor LaOFeAs
DMFT results for Hamiltonian
and Coulomb interaction
parameters calculated
with Wannier functions
for Fe3d bands only
U=0.8 eV
J=0.5 eV

16. Novel superconductor LaOFeAs
DMFT results for Hamiltonian
and Coulomb interaction
parameters calculated
with Wannier functions
for all bands (O2p,As4p,Fe3d)
U=3.5 eV
J=0.8 eV
Moderately correlated
regime with significant
renormalization for
electronic states on the
Fermi level (effective mass m*~2) but
no Hubbard band.

17. Novel superconductor LaOFeAs spectra
Comparison of calculation results with experimental spectra confirms moderately correlated regime without Hubbard band.
V.I. Anisimov, E.Z. Kurmaev, A. Moewes, I.A. Izyumov, Physica C 469, 442–447 (2009)

18. M.Rotter et al. (2008) tetragonal structure I4/mmm (139)
BaFe2As2: parent compound
Critical temperatures:
- stochiometric under 40 kbar Tc=29 K P.L. Alireza et al. (2009)
- doped Ba1-xKxFe2As2 Tc=38 K M.Rotter et al. (2008)
M.Rotter et al. (2008)
tetragonal structure
I4/mmm (139)
Evidences for correlation effects in pnictides:
- ARPES measurements: bands narrowing comparing with LDA bands ~ 2 times
- dHvA experiments: electronic mass enhancement 1.7÷2.1

19. No Hubbard bands Moderate renormalization
BaFe2As2: LDA vs DMFT and m* estimation
DMFT spectral functions:
No Hubbard bands
Moderate renormalization
Quantitative estimation of the correlation strength:
Orbitals
3dxy
3dyz, xz
3d3z2-r2
3dx2-y2
m*/m
2.06
2.07
2.05
1.83
S. Skornyakov et al, Phys. Rev. B 80, (2009)

20. BaFe2As2: Hubbard bands or hybridization?
Stripes: lines
with the LDA 3d band width
Effects of As p – Fe d hybridization

21. BaFe2As2: DMFT results vs ARPES experiment
Chang Liu et al. (2008) This work
- Good agreement with PES and
ARPES data
- DMFT bands εDMFT(k) are
very well represented by scaling
εDMFT(k)=εLDA(k)/(m*/m)
S. Skornyakov et al, Phys. Rev. B 80, (2009)
S. de Jong et al. (2009)

22. BaFe2As2: correlation strength
SrVO3
m*/m=2
Substantial spectral weight transfer
from the quasiparticle states to well pronounced Hubbard bands
m*/m=2
No spectral weight transfer
from the quasiparticle states to
Hubbard bands

23. LaOFeP: correlation strength
Transition temperature Tc ~ 4 K in LaOFeP
in contrast to Tc ~ 26–55 K in
RO1−xFxFeAs (R = La, Sm)
Correlation effects in LaOFeP are comparable with LaOFeAs and BaFe2As2
m*~2
S. Skornyakov et al, Phys. Rev. B 81, (2010)

24. LaOFeP: DMFT results vs ARPES experiment
-Good agreement with experiment
(overall shape, size and position of electron and hole pockets)
-Band narrowing corresponding to
m*/m~2 (in comparison with LDA) for all orbitals, like in other pnictides
-No obvious connection between correlation strength and superconductivity in pnictide superconductors

25. Magnetic properties of pnictides
For nonsuperconducting pnictides antiferromagnetic spin density wave is observed in FeAs layers (TN=140K for LaFeAsO)
Anomalous χ(T) at T>Tc, T>TN
non Pauli and non Curie-Weiss type1,2
Linear increase – features:
- Slope of χ(T) curve is doping-independent
- universal property of paramagnetic phase
in all pnictides, superconducting or not
Attempts to explain due to inter-site
magnetic correlations3
1Klingeler et al. EPL (2009), 2Zhang et al. PRB (2010), 3Korshunov et al. PRL PRL (2009) 25

26. LaFeAsO1-xFx – first discovered pnictide superconductor Tc=26 K
Klingeler et al. PRB (2010) 26

27. LDA+DMFT: LaFeAsO spectral functions27

28. LaFeAsO: magnetic susceptibility calculations results
Taking into account local correlations in DMFT is enough to obtain linear increase in temperature dependence of χ(T) !
Contributions χi(T) are orbital dependent
What is possible mechanism for linear increase in χi(T)?
S.L. Skornyakov, A.A. Katanin, V.I. Anisimov PRL 106, (2011) 28

29. LaFeAsO: magnetic susceptibility calculations results
Qualitatively χi(T) temperature dependence is defined by one-electron spectra obtained in LDA+DMFT calculations 29

30. LaFeAsO: magnetic susceptibility calculations results
Increase of χ(T) for x2-y2 is provided by peculiarities of the other orbitals
Temperature
387 K
580 K
1160 K
ImΣxy(EF)
-0.142
-0.242
-0.454
ImΣyz,xz(EF)
-0.131
-0.163
-0.306
ImΣ3z2-r2(EF)
-0.054
-0.092
-0.228
ImΣx2-y2(EF)
-0.053
-0.101
-0.334 30

31. BaFe2As2: spectral properties from LDA+DMFT

32. BaFe2As2: magnetic susceptibility calculations results32

33. BaFe2As2: uniform susceptibility and single-particle properties33

34. Peaks near Fermi in some superconductors
Borisenko et al
PRL 105, (2010)
LiFeAs
BCFA
PCCO
LSCO
Sr2RuO4
OD-YBCO
OD-BSCCO
BKFA
© S.V. Borisenko

35. LiFeAs: k-resolved spectrum from LDA+DMFT35

36. LiFeAs: k-resolved spectrum from LDA+DMFT36

37. LiFeAs: k-resolved spectrum from LDA+DMFT37

38. Comparison of calculated and experimental spectra for LiFeAs
Borisenko et al. PRL 105, (2010)

39. Conclusion
Dynamical Mean-Field approach combined with DFT methods – powerful tool for material-specific investigation
Wannier functions – convenient and illustrative basis making LDA+DMFT scheme numerically feasible
LDA+DMFT results for LaOFeAs, BaFe2As2 ,, LiFeAs and LaOFeP are in good agreement with PES and ARPES data
Calculated quasiparticle bands renormalization corresponding to effective mass enhancement m*/m~2~3 observed simultaneously with the absence of Hubbard bands shows pnictide superconductors as moderately correlated systems but far from metal-insulator Mott transtion
Linear increase with temperature for uniform magnetic susceptibility observed experimentally is successfully reproduced in LDA+DMFT calculations.

40. Wannier functions: projection technique
Wei Ku et al. (2002): A good approximation to Maximally localized Wanner functions is projection of trial orbitals onto the subspace of Bloch functions
In our case = site centered pseudoatomic orbitals