coulomb correlation effects in electronic structure of iron pnictide...

Coulomb correlation effects in electronic structure of iron pnictide

superconductors

Vladimir I. Anisimov

Institute of Metal PhysicsEkaterinburg, Russia

Dynamical Mean-Field Theory (DMFT)

Combining DMFT with Density Functional Theory methods (LDA+DMFT)

Realization of the LDA+DMFT computational scheme inWannier functions basis

Pnictide superconductors LaOFeAs, BaFe2As2 LiFeAs and LaOFeP investigated within LDA+DMFT method

Summary

Outline

Object of investigation: interacting lattice fermions

Simplest description – Hubbard model

Unsolvable problem for d≥2Reason – correlation phenomena

Square lattice, z=4

†

, ,i j i j

i j i

H t c c U n nσ

↑ ↓< >

= − +∑ ∑

i j i jn n n n≠

Approximations need to be made

Dynamical Mean-Field Theory

Metzner, Vollhardt (1989)d→∞

Georges, Kotliar (1992)Jarrell (1992)

mapping onto impurity problem, self-consistent equations

Real latticeEffective impurity problem

Mapping

Dynamical Mean-Field Theory

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:

1 10 ( ) ( ) ( )n n nG i G i iω ω ω− −= +Σ

1 10( ) ( ) ( )n n new ni G i G iω ω ω− −Σ = −

( )niωΣ

Impurity solver

0G

newG( )scf niωΣ

Self-consistencycheck

( )( )( )n

n n

NG i di i

εω εω µ ω ε

=+ −Σ −∫

Start: noninteracting density of states N(ε), initial guess for Σ(iωn)

+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 (O2p orbitals)

LDA+DMFT computational scheme

1 10( ) ( ) ( )n n new ni G i G iω ω ω− −Σ = −

Effective Hamiltonianconstruction

1( ) ( ) ( )n n nG i dk i H k iω ω µ ω

−⎡ ⎤= + − −Σ⎣ ⎦∫r r

1 10 ( ) ( ) ( )n n nG i G i iω ω ω− −= +Σ

( )H kr

Impurity solver

0G

newG

Self-consistencycheck

( )niωΣ

( )scf niωΣ

Hilbert transform N(ε)→G(iωn) can not be applied to

charge transfer insulators

Solution – use full Hamiltonian instead of N(ε) ( )H kr

LDA+DMFT computational scheme

Possible ways to define material-specific : ( )H kr

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 functionsN. Marzari and D. Vanderbilt (1997), F. Lechermann et al (2006)

(ii) Atomic-orbitals projected Wannier functions ina) LMTO basis set Anisimov et. al. (2005)b) Pseudopotential basis set

Trimarchi et. al. (2008), Dm. Korotin et. al. (2008)

Effective Hamiltonian construction

1T ikTn nk

k

W e−= ΨΩ∑

rr r

rr

T ikTn nk

k

W W e−= ∑rr r

rr

% %

2

1 1 1( )i

N

nk ik ik nk ik ik nki N E k E

Wε

φ φ= ≤ ≤

= Ψ Ψ = Ψ Ψ∑ ∑r r r r r r rr

% % % % %

( )nqnkq

W k k qω= −∑r rr

r r r% %

Wannier functions :TnWr

Wei Ku et al. (2002): A good approximation to Maximally localized Wanner functions is projection of trial orbitals onto the subspace of Bloch functions

TnWr

% Tnφr

In our case = site centered pseudoatomic orbitals Tnφr

Wannier functions: projection technique

2 2

1 1

* *( ) ( ) ( ) ( ) ( )N N

WFnm i in im ink ik ik mk

i N i N

H k W k W b k b k kε ε= =

⎛ ⎞= Ψ Ψ =⎜ ⎟

⎝ ⎠∑ ∑r r r r

r r r r r

2

1

( )( ) ( ) ( )N

WF T T WF ik T Tnm n i m nmik ik

i Nk k

H T T W k W H k eε ′ ′−

=

⎛ ⎞′ − = Ψ Ψ =⎜ ⎟

⎝ ⎠∑∑ ∑

rr r r r

r rr r

r rr r

( ) ( )2

1

0 0 *( ) ( ) ( ) ( )N

WFnm n i F m in im i Fik ik

i Nk k

Q W k E W b k b k k Eθ ε θ ε=

⎛ ⎞= Ψ − Ψ = −⎜ ⎟

⎝ ⎠∑∑ ∑r rr r

r r r r

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

1 ( 1),2

WFDFT d d d d dd

d

E U n n n Q= − =∑ DFTd

d

EUn

∂=

∂

Wannier functions: applications

LDA calculation – band structureOrbitals of interest choice (interacting d- or f-orbitals) for projectionEffective Hamiltonian construction for Wannier functions

LDA Effective Hamiltonian

projection

Interaction parameters U and J calculation in constrain DFTDMFT solution of the problem defined by Hamiltonian

†

,

1ˆ ˆ ˆ ˆ ˆ( )2

WFnm nmnT mT nT mT

nmTT n m T

H H T T c c U n n′ ≠

′= − +∑ ∑r r r rr r r

r r

( )WFnmH T T′ −

r r

LDA+DMFT scheme in Wannier functions basis

dxz-like Wannier function modulus square isosurface:

Wannier states constructed for different energy intervals (Dm.Korotin et al. (2008)):

L G X W L K G

-8

-6

-4

-2

0

Ene

rgy

(eV

)

L G X W L K G

-8

-6

-4

-2

0

Ene

rgy

(eV

)

2.5; 1.5⎡ ⎤−⎣ ⎦ 2.5; 1.5⎡ ⎤−⎣ ⎦

Wannier functions: NiO example

eVeV

Novel superconductor LaOFeAs

Tc=26K for F content ~11%

Y. Kanamura et al. J. Am. Chem. Soc. 130, 3296 (2008)


02468

10Total

-5 -4 -3 -2 -1 0 1 2Energy, eV, EF=0

02d-Fe

02

Den

sity

of s

tate

s, (

eV. a

tom

)-1

p-As02p-O

d (x2-y2) Wannier functions (WF) calculatedfor all bands (O2p,As4p,Fe3d) andfor Fe3d bands only

All bands WFconstrain DFTU=3.5 eVJ=0.8 eV

Fe3d band only WFconstrain DFTU=0.8 eVJ=0.5 eV

V.Anisimov et al, J. Phys.: Condens. Matter 21, 075602 (2009)


0

0,5

Spe

ctra

l fun

ctio

n, e

V-1

xy

0

0,5 yz, zx

0

0,5 3z2-r

2

-3 -2 -1 0 1 2 3Energy, eV, EF=0

0

0,5 x2-y

2

DMFT results for Hamiltonianand Coulomb interactionparameters calculated with Wannier functionsfor Fe3d bands onlyU=0.8 eVJ=0.5 eV


DMFT results for Hamiltonianand Coulomb interactionparameters calculated with Wannier functionsfor all bands (O2p,As4p,Fe3d)U=3.5 eVJ=0.8 eV

Moderately correlated regime with significant renormalization for electronic states on the Fermi level (effective mass m*~2) butno Hubbard band.

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)

M.Rotter et al. (2008)tetragonal structure

I4/mmm (139)

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)

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

BaFe2As2: parent compound

*0 0

Im Re/ 1 1( )

| |i im mi ω ωω ω= =

∂ Σ ∂ Σ= − = −

∂ ∂

( ) ( )Padeiω ωΣ ⎯⎯⎯→Σ

( ) ( )( ) 11 ( )k

A H kω ω µ ωπ

−= − + − −Σ∑r

r

Orbitals 3dxy 3dyz, xz 3d3z2-r

2 3dx2-y

2

m*/m 2.06 2.07 2.05 1.83

No Hubbard bandsModerate renormalization

Quantitative estimation of the correlation strength:

DMFT spectral functions:

BaFe2As2: LDA vs DMFT and m* estimation

S. Skornyakov et al, Phys. Rev. B 80, 092501 (2009)

BaFe2As2: Hubbard bands or hybridization?

Effects of As p – Fe d hybridization

Stripes: lineswith the LDA 3d band width

dcHω µ+ +

- Good agreement with PES andARPES data

- DMFT bands εDMFT(k) arevery well represented by scaling

εDMFT(k)=εLDA(k)/(m*/m)

S. de Jong et al. (2009)

Chang Liu et al. (2008) This work

BaFe2As2: DMFT results vs ARPES experiment


BaFe2As2: correlation strength BaFe2As2 SrVO3

m*/m=2Substantial spectral weight transfer from the quasiparticle states to well

pronounced Hubbard bands

m*/m=2No spectral weight transfer

from the quasiparticle states toHubbard bands

LaOFeP: correlation strength

Transition temperature Tc ~ 4 K in LaOFePin contrast to Tc ~ 26–55 K in RO1−xFxFeAs (R = La, Sm)

Correlation effects in LaOFeP are comparable with LaOFeAs and BaFe2As2m*~2


LaOFeP: DMFT results vs ARPES experiment

-Good agreement with experiment(overall shape, size and position of electron and hole pockets)

-Band narrowing corresponding tom*/m~2 (in comparison with LDA) for all orbitals, like in other pnictides

-No obvious connection between correlation strength and superconductivity in pnictide superconductors

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

Magnetic properties of pnictides

1Klingeler et al. EPL 86 37006 (2009), 2Zhang et al. PRB 81 024506 (2010), 3Korshunov et al. PRL PRL 102 236403 (2009)

Klingeler et al. PRB 81 024506 (2010)

LaFeAsO

LaFeAsO1-xFx – first discovered pnictide superconductor Tc=26 K

LDA+DMFT: LaFeAsO spectral functions

*0 0

Im Re/ 1 1 2.4 3.0( )

| |im mi ω ωω ω= =

∂ Σ ∂ Σ= − = − = ÷

∂ ∂

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)?

( )( ) ( )( )

i ii

ih h

n T n TM TTE E

σ σχ −∂ −∂

= =∂ ∂

( ) ( )

h

M TTE

χ ∂=

∂

S.L. Skornyakov, A.A. Katanin, V.I. Anisimov PRL 106, 047007 (2011)


Qualitatively χi(T) temperature dependence is defined by one-electron spectra obtained in LDA+DMFT calculations

( )0

,

1 ( , ) ( , )m mn nmnk i

T G k i G k iω

χ ω ωβ

= − ∑∑r

r r


( )0

,

1 ( , ) ( , )mn mn nmk i

T G k i G k iω

χ ω ωβ

= − ∑rr r ( ) ( )0 0

m mnn

T Tχ χ=∑

Temperature 387 K 580 K 1160 K

ImΣxy(EF) -0.142 -0.242 -0.454ImΣyz,xz(EF) -0.131 -0.163 -0.306ImΣ3z2-r2(EF) -0.054 -0.092 -0.228ImΣx2-y2(EF) -0.053 -0.101 -0.334

Increase of χ(T) for x2-y2 is provided by peculiarities of the other orbitals

BaFe2As2: spectral properties from LDA+DMFT

*0 0

Im Re/ 1 1 2.73 3.74( )


∂ Σ ∂ Σ= − = − = ÷

∂ ∂

BaFe2As2: magnetic susceptibility calculations results

( )( ) ( )( )

i ii

ih h

n T n TM TTE E

σ σχ −∂ −∂

= =∂ ∂

( ) ( )

h

M TTE

χ ∂=

∂

BaFe2As2: uniform susceptibility and single-particle properties

( )0

,

1 ( , ) ( , )m mn nmnk i

T G k i G k iω

χ ω ωβ

= − ∑∑r

r r

Peaks near Fermi in some superconductors

OD-BSCCOOD-YBCOSr2RuO4

LiFeAs PCCO LSCOBCFA

BKFA

© S.V. Borisenko

Borisenko et alPRL 105, 067002 (2010)

LiFeAs: k-resolved spectrum from LDA+DMFT

*0 0

Im Re/ 1 1 2.00 3.75( )


∂ Σ ∂ Σ= − = − = ÷

∂ ∂

LiFeAs: k-resolved spectrum from LDA+DMFT

Comparison of calculated and experimental spectra for LiFeAs

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 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.

1T ikTn nk

k

W e−= ΨΩ∑

rr r

rr

T ikTn nk

k

W W e−= ∑rr r

rr

% %

2

1 1 1( )i

N

nk ik ik nk ik ik nki N E k E

Wε

φ φ= ≤ ≤

= Ψ Ψ = Ψ Ψ∑ ∑r r r r r r rr

% % % % %

( )iqikq

c k k qΨ = −∑r rr

r r r ( )nqnkq

a k k qφ ′′

′= −∑r rr

r r r

( )in nqnk iki q

W b k k qω= Ψ = −∑ ∑r r rr

r r r%% %

*( ) ( ) ( )in iq iqq

b k c k a k′ ′′

≡ ∑ r rr

r r r%

Wannier functions :TnWr

Wei Ku et al. (2002): A good approximation to Maximally localized Wanner functions is projection of trial orbitals onto the subspace of Bloch functions

TnWr

% Tnφr

In our case = site centered pseudoatomic orbitals Tnφr

Wannier functions: projection technique

2

1

( ) ( ) ( )N

nq in iqi N

k b k c kω=

≡ ∑r r

r r r%

coulomb correlation effects in electronic structure of iron pnictide...

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