Coulomb correlation effects in electronic structure of iron

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

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 La. OFe. As, Ba. Fe 2 As 2 Li. Fe. As and La. OFe. P investigated within LDA+DMFT method Summary

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

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

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:

LDA+DMFT computational scheme Start: noninteracting density of states N(ε), initial guess for Σ(iωn) Self-consistency check Impurity solver + Can be applied for Mott insulators with N(ε) – LDA DOS of d-band – Restricted to single-orbital or degenerate orbitals case Can not be applied if orbitals of interest (d-orbitals) strongly hybridize with other electronic states (O 2 p orbitals)

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

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)

Wannier functions: projection technique Wannier functions : 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

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

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

Wannier functions: Ni. O example Wannier states constructed for different energy intervals (Dm. Korotin et al. (2008)): e. V dxz-like Wannier function modulus square isosurface: e. V

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

Novel superconductor La. OFe. As All bands WF constrain DFT U=3. 5 e. V J=0. 8 e. V d (x 2 -y 2) Wannier functions (WF) calculated for all bands (O 2 p, As 4 p, Fe 3 d) and for Fe 3 d bands only V. Anisimov et al, J. Phys. : Condens. Matter 21, 075602 (2009) Fe 3 d band only WF constrain DFT U=0. 8 e. V J=0. 5 e. V

Novel superconductor La. OFe. As DMFT results for Hamiltonian and Coulomb interaction parameters calculated with Wannier functions for Fe 3 d bands only U=0. 8 e. V J=0. 5 e. V

Novel superconductor La. OFe. As DMFT results for Hamiltonian and Coulomb interaction parameters calculated with Wannier functions for all bands (O 2 p, As 4 p, Fe 3 d) U=3. 5 e. V J=0. 8 e. V Moderately correlated regime with significant renormalization for electronic states on the Fermi level (effective mass m*~2) but no Hubbard band.

Novel superconductor La. OFe. As 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)

Ba. Fe 2 As 2: parent compound Critical temperatures: tetragonal structure - stochiometric under 40 kbar Tc=29 K P. L. Alireza et al. (2009) M. Rotter et al. (2008) I 4/mmm (139) - doped Ba 1 -x. Kx. Fe 2 As 2 Tc=38 K M. Rotter et al. (2008) Evidences for correlation effects in pnictides: - ARPES measurements: bands narrowing comparing with LDA bands ~ 2 times - d. Hv. A experiments: electronic mass enhancement 1. 7÷ 2. 1

Ba. Fe 2 As 2: LDA vs DMFT and m* estimation DMFT spectral functions: No Hubbard bands Moderate renormalization Quantitative estimation of the correlation strength: Orbitals 3 dxy 3 dyz, xz 3 d 3 z 2 -r 2 3 dx 2 -y 2 m*/m 2. 06 2. 07 2. 05 1. 83 S. Skornyakov et al, Phys. Rev. B 80, 092501 (2009)

Ba. Fe 2 As 2: Hubbard bands or hybridization? Effects of As p – Fe d hybridization Stripes: lines with the LDA 3 d band width

Ba. Fe 2 As 2: 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. de Jong et al. (2009) S. Skornyakov et al, Phys. Rev. B 80, 092501 (2009)

Ba. Fe 2 As 2: correlation strength Ba. Fe 2 As 2 m*/m=2 No spectral weight transfer from the quasiparticle states to Hubbard bands Sr. VO 3 m*/m=2 Substantial spectral weight transfer from the quasiparticle states to well pronounced Hubbard bands

La. OFe. P: correlation strength Transition temperature Tc ~ 4 K in La. OFe. P in contrast to Tc ~ 26– 55 K in RO 1−x. Fe. As (R = La, Sm) Correlation effects in La. OFe. P are comparable with La. OFe. As and Ba. Fe 2 As 2 m*~2 S. Skornyakov et al, Phys. Rev. B 81, 174522 (2010)

La. OFe. P: 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

Magnetic properties of pnictides For nonsuperconducting pnictides antiferromagnetic spin density wave is observed 1 Klingeler in Fe. As layers (TN=140 K for La. Fe. As. O) Anomalous χ(T) at T>Tc, T>TN non Pauli and non Curie-Weiss type 1, 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 correlations 3 et al. EPL 86 37006 (2009), 2 Zhang et al. PRB 81 024506 (2010), 3 Korshunov et al. PRL 102 236403 (2009)

La. Fe. As. O 1 -x. Fx – first discovered pnictide superconductor Tc=26 K Klingeler et al. PRB 81 024506 (2010)

LDA+DMFT: La. Fe. As. O spectral functions

La. Fe. As. O: 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, 047007 (2011)

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

La. Fe. As. O: magnetic susceptibility calculations results Increase of χ(T) for x 2 -y 2 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 ImΣ 3 z 2 -r 2(EF) -0. 054 -0. 092 -0. 228 ImΣx 2 -y 2(EF) -0. 053 -0. 101 -0. 334

Ba. Fe 2 As 2: spectral properties from LDA+DMFT

Ba. Fe 2 As 2: magnetic susceptibility calculations results

Ba. Fe 2 As 2: uniform susceptibility and single-particle properties

Peaks near Fermi in some superconductors Borisenko et al PRL 105, 067002 (2010) Li. Fe. As BCFA Sr 2 Ru. O 4 OD-YBCO PCCO OD-BSCCO LSCO BKFA © S. V. Borisenko

Li. Fe. As: k-resolved spectrum from LDA+DMFT

Li. Fe. As: k-resolved spectrum from LDA+DMFT

Li. Fe. As: k-resolved spectrum from LDA+DMFT

Comparison of calculated and experimental spectra for Li. Fe. As Borisenko et al. PRL 105, 067002 (2010)

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 La. OFe. As, Ba. Fe 2 As 2 , , Li. Fe. As and La. OFe. P 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.

Wannier functions: projection technique Wannier functions : 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