Effects of electronic correlations in iron and iron pnictides

Effects of electronic correlations in

iron and iron pnictidesA. A. Katanin

In collaboration with:A. Poteryaev, P. Igoshev, A. Efremov,

S. Skornyakov, V. Anisimov

Institute of Metal Physics, Ekaterinburg, Russia

Special thanks to Yu. N. Gornostyrev for stimulating discussions

Iron properties

Mikhaylushkin, PRL 99, 165505 (2007)

Arajs. J.Appl.Phys. 31, 986 (1960)Parsons, Phil.Mag. 3, 1174 (1959)

a-iron: TC = 1043K, meff =3.13mB

a g Ts =1185 Ka - bcc, g - fcc, e - hcp g-iron: qCW =-3450K, meff =7.47mB

• Itinerant approach (Stoner theory)𝐼𝑁 (𝐸𝐹 )=1

Large DOS implies ferromagnetism, provided that other magnetic or charge instabilities are less important- Too large magnetic transition temperatures, no CW-law

• Moriya theory: paramagnons Reasonable magnetic transition temperatures, CW law

• Local moment approaches (e.g. Heisenberg model) CW law

Rhodes-Wollfarth diagram

a-Iron (almost) fulfills Rhodes-Wollfarth criterion(pc/ps 1)

Proposals for iron:• Local moments are formed by eg electrons (Goodenough, 1960)• 95% d-electron localization (Stearns, 1973)• Local moments are formed from the vH singularity eg states

(Irkhin, Katsnelson, Trefilov, 1993)

The magnetism of iron

Local moments (Heisenberg model)

• Can one decide unbiasely (ab-initio), which states are localized (if any) ?

• What is the correct physical picture for decribing local magnetic moments in an itinerant system? Itinerant (Stoner and Moriya

theory)

Mixed (Shubin s-d(f)

= FM Kondo model)

How the local moments (if they exist) influence magnetic properties? What is the similarity and differences between magnetism of a- and g- iron?

a-Iron shows features of both, itinerant (fractional magnetic moment) and localized (Curie-Weiss law with large Curie constant) systems

Dynamical Mean Field Theory

The self-energy of the embedded atom coincides with that of the solid (lattice model), which is approximated as a k-independent quantity

Energy-dependent effective medium theory

A. Georges et al., RMP 68, 13 (1996)

( )

1 1( ) ( ) ( )locG - - = -

k k

1( )( )locG

e m =

- -

Spin-polarized LDA+DMFT

Lichtenstein, Katsnelson, Kotliar, PRL 87, 67205 (2001)

U = 2.3 eV, J = 0.9 eV

Magnetic moment 3.09 (3.13)Critical temperature 1900 K (1043K)

a (Bcc) iron: band structure

t2g и eg states are qualitatively different and weakly hybridized

Correlations can “decide”,which of them become local

eg t2g

A. Katanin et al., PRB 81, 045117 (2010)

a-iron: orbitally-resolved self-energy

Imaginary frequencies

t2g states - quasi-particleseg states - non-quasiparticle! Bulla et al., PRB 64, 45103 (2001)

Linear for the Fermi liquidDivergent for an insulator

Comparison to MIT:

A. Katanin et al., PRB 81, 045117 (2010)

Real frequencies

From: Bulla et al., PRB 64, 45103 (2001)

Self-energy and spectral functions at the real frequency axis

Comparison to MIT:

a-Fe

How to see local moments:local spin correlation function

J=0.9 J=0

S(0

)S(

)

Local moments are stable when

( ) (0)z zS S const

Fulfilled at the conventional Mott transition. Can it be fulfilled in the metallic phase ?

A. Katanin et al., PRB 81, 045117 (2010)

Fourier transform of spin correlation function

2

( ) ( / )3

eff f TT

m =

Fourier transform of spin correlation function

2

( ) ( / )3

eff f TT

m =

Local moments formed out of eg states do exist in iron!

Which form of one can expect for the system with local moments?

(0) ( ) const( )S S 2

0,0( ) (0) ( ) / 3ni

nn z zi d S S e S T

=

2

( )3 | |

effn

n

iT

m g g

=

2 2

2 2Re ( )3

eff

Tm g

g=

2

( )3

eff iT i

m g g

=

g is the damping of local collective excitations

2

2 2Im ( )3

eff

Tm g

g=

Broaden delta-symbol:

𝜇eff❑ =3.3𝜇𝐵

𝛾≈𝑇 /2

For a-iron:

(𝜔≪ 𝐽 )

p(eg) = 0.56p(t2g) = 0.45p(total)=1.22

Curie law for local susceptibility

2 2 ( 1) / (3 )Bg p p T m=

agrees with the experimental data (known also after A.Liechtenstein, M. Katsnelson, and G. Kotliar, PRL 2001)

local moment

eg

t2g

Total

Effective model

The local moments are coupled via RKKY-type of exchange:

2 2

2 2, , ,' '

( ) 22g g g g

g g

deff e t t e i im i im

i m t i m t

JH H H H U N n J

-

= - - S s

RKKY type(similar to s-d Shubin-Vonsovskii model).

The theoretical approaches, similar to those for s-d model can be used

g-(fcc) iron

Which physical picture (local moment, itinerant) is suitable to describe g-iron ?

What is the prefered magnetic state for the g iron at low T (and why)?

TN≈100K

LDA DOS

The peak in eg band is shifted by 0.5eV downwards with respect to the Fermi level

g-(fcc) iron

More itinerant than a-iron ?

P. A. Igoshev et al., PRB 88, 155120 (2013)

DOS with correlations

Static local susceptibility

P. A. Igoshev, A. Efremov, A. Poteryaev, A. K., and V. Anisimov,PRB 88, 155120 (2013)

Dynamic local susceptibility

Size of local moment

Magnetic state: Itinerant picture

Comparison of energies in LDA approachShallcross et al., PRB 73, 104443 (2006)

QX=(0,0,2) SDW2

Magnetic state: Heisenberg model picture

Heisenberg model

For stability of (0,0,2) state one needs J1>0, J2<0.

A. N. Ignatenko, A.A. Katanin, V.Yu.Irkhin, JETP Letters 87, 555 (2008)

The polarization bubble, low T

m m'k

k+q

2(1,0,0) 2(1/2,1/2,1/2) 2(1,1/2,0)

T=290К

LDA

LDA+DMFT

Experimental magnetic structure

Tsunoda, J.Phys.: Cond.Matt. 1, 10427 (1989)Naono and Tsunoda, J.Phys.: Cond.Matt. 16, 7723 (2004)

q = (2/a) (1, 0.127, 0)

Fermi surface nesting

(0,x,2) state is supported by the Fermi surface geometry – an evidence for itinerant nature of magnetism

Colorcoding: red – eg, green – t2g, blue – s+p

The polarization bubble, high TLDA

LDA+DMFT

T=1290К

Uniform susceptibility

From high-temperature part:

1/

m m'k

k

g-(fcc) iron

The experimental value of the Curie constant is reproduced by the theory, although the absolute value of paramagnetic Curie temperature appears too large

exp

DMFT

1500

2700100 (small particles)N

K

KT K

-

-

Strong frustration! Nonlocal correlations are important

𝜇𝐶𝑊=7.7𝜇𝐵

Magnetic exchange in g-iron 𝜒𝐪=

𝜒 0

1 − 𝐽𝐪 𝜒0𝜒𝐪=

𝜒 irr (𝐪 )1 − Γ 𝜒 irr(𝐪)

𝐽𝐪=− ¿¿¿𝐽𝟎=−2500 𝐾𝐽𝐐=1200𝐾

The Neel temperature is much larger than the experimental one,similar to the result of the Stoner theory:

o Paramagnonso Frustration, i.e. degeneracy of spin susceptibility in different directions

# 1 2 3 4 5 6 7 8J z/2,K

-669 173 -449 17 -25 -123

-116 29

Local spin susceptibility of Ni

A. S. Belozerov, I. A. Leonov, and V. I. Anisimov, PRB 2013

Iron pnictide LaFeAsO Antiferromagnetic fluctuations Superconductivity Itinerant system in the normal state

Effect of electronic correlations?

Possibility of local moment formation?

Density of states

Damped qp states

qp states

No qp states

Elec

troni

c co

rrela

tions

387K 580К 1160К

xy 0.142 0.242 0.454xz, yz 0.131 0.163 0.3063z2-r2 0.054 0.092 0.228x2-y2 0.053 0.101 0.334

dxz, dyz, dxy states can bemore localized

Local susceptibility387K 580К 1160

Кxy -0.142 -0.242 -0.454xz, yz -0.131 -0.163 -0.3063z2-r2 -0.054 -0.092 -0.228x2-y2 -0.053 -0.101 -0.334

Spin correlation functions

The situation is similar to g-iron, i.e.local moments may exist

at large T only, and, therefore,seem to have no effect on superconductivity

Orbital-selective uniform susceptibility

Local fluctuations are responsiblefor the part of linear-dependentterm in (T)

S. L. Skornyakov, A. Katanin, and V. I. Anisimov, PRL ’ 2011

Summary

The existence of local moments is observed within the LDA+DMFT approach

The formation of local moments is governed by Hund interaction

In alfa-iron:

The peculiarities of electronic properties (flat bands, peaks of density of states)near the FL may lead to the formation of local moments;

Analysis of orbitally-resolved static and dynamic local susceptibilitiesproves to be helpful in classification of different substances regarding the degree of local moment formation

Local moments are formed at high T>1000K, where this substance exist in nature, but not at low-T (in contrast to alfa-iron); the low-temperature magnetism appears to be more itinerant

Antiferromagnetism is provided by nesting of the Fermi surface

In gamma-iron:

Conclusions

Electronic correlations are important, but, similarly to g-iron, local moments may be formed at large T only

Different orbitals give diverse contribution to magneticproperties

Linear behavior of uniform susceptibility is (at least partly) due to peaks of density of states near the Fermi level

In the iron pnictide:

Thank you for attention !

Spin correlation functions

Spectral functions

Damped qp states

qp states

No qp states

Effective model and diagram technique

2

2

2

,

, , ,

2

( )2

g g

g

g g

deff t e i im

i m t

im imi m e m t

H H H I

IU n n

= -

-

S s

Treat eg electrons within DMFT and t2g electrons perturbatively Simplest way is to decouple an interaction and integrate out t2g electrons

1 2 3 1 1 2 2 3 3

1 2 3 1 2 3

1

,

,,

[ ( 2 )( 2 )]

( 2 ) ( 2 ) ( 2 )

( 2 ) ...

g

i i

m m mm m me q q q q q q qmm

q mm

mm m m m m mq q q abcd q q a q q b q q c

q m

mq q q q q q

L L R I I

I I I

I

-- - -

- - - - - -

= -

t t t

t t t

t

t S S

S S S

S

“bare” quadratic term

quartic interaction

(similar to s-d Shubin-Vonsovskii model).

mmq

=

1 2 3 ,mm m mq q q abcd

=

0, gq e =

1 2 3 1 2 3 1 2 3

(4),, g

abcd a b c dq q q q q q q q q c eS S S S- - - = =

Diagram technique: perspective

The dynamic susceptibility

2

2 2

2

,

,

10 1 0 2 0,

0 1 2 2 (4) 0,

( )

( ) 4 2

2 ( ) 4 4

g

g

g g g

g g

q t q

q q e

q t t e q

q q e q t

R I R R

R

I I

I I I

--

-

- = - -

RKKY

“Moriya”correction

Influence of itinerantelectrons on local momentdegrees of freedom

bare

bare

Exchange integrals and magnetic properties can be extracted

• Two different approaches to magnetism of transition metals(and explaining Curie-Weiss behavior):

- Itinerant (Stoner, Moriya, …)- Local moment (Heisenberg, …)

Can one unify these approaches(one band: Moriya, degenerate bands: Hubbard, …)

More importantly: what is the ‘adequate’ (‘appropriate’) effective model, describing magnetic properties of transition metals ?

Since they are (good) metals, at first glance no ‘true’local moments are formed

However, under some conditions the formationof (orbital-selective) local moments is possible:

- Weak hybridization between different states (e.g.t2g and eg)

- Presence of Hund exchange interaction

- Specific shape of the density of states

Local moments in transition metals

Since they are (good) metals, at first glance no ‘true’local moments are formed

However, under some conditions the formationof (orbital-selective) local moments is possible:

- Weak hybridization between different states (e.g.t2g and eg)

- Presence of Hund exchange interaction

- Specific shape of the density of states

Local moments in transition metals

Dependence on imaginary frequency

Paramagnetic LDA+DMFTU = 2.3 eV, J = 0.9 eV, T = 1120 K

t2g states eg states

Weakly correlated compound ?!?!?!?

t2g и eg состояния качественно различны и слабо гибридизованы

Важно учесть влияние электронных корреляций

U dependence

J = 0.9 eV, = 10 eV-1

Stability with temperature

Weak itinerant magnets Saturation magnetic moment is small The thermodynamic properties are detrmined by paramagnons;

Hertz-Moriya-Millis theory: for ferromagnets (d=3, z=3) the bosonic mean-field (Moriya) theory is sufficient to describe qualitatively thermodynamic properties even close to QCP.

0, 4/3

0 01...3 , ,1

n

n n

k iabab

b i k i

T TU

=

-

k

Curie-Weiss-like susceptibility

“paramagnon”0,0,1

n

n

k i

k iU

-

0abab

Frustration in Heisenberg FCC model

Polarization bubble

m m'eg

t2gt2g-e2g

G. Stollhoff, 2007

mmq

=

1 2 3 ,mm m mq q q abcd

=

0, gq e =

1 2 3 1 2 3 1 2 3

(4),, g

abcd a b c dq q q q q q q q q c eS S S S- - - = =

Diagram technique: perspective

Spin correlation function at different U

S(0

)S(

)

almost flat !eg

t2g

Weak itinerant magnets Saturation magnetic moment is small The thermodynamic properties are detrmined by paramagnons;

Hertz-Moriya-Millis theory: for ferromagnets (d=3, z=3) the bosonic mean-field (Moriya) theory is sufficient to describe qualitatively thermodynamic properties even close to QCP.

0, 4/3

0 01...3 , ,1

n

n n

k iabab

b i k i

T TU

=

-

k

Curie-Weiss-like susceptibility

“paramagnon”0,0,1

n

n

k i

k iU

-

0abab

Effective model

2

2

2

,

, ,

2

( )2

g g

g

g

deff t e i im

i m t

i imi m t

H H H I

IU N n

= -

-

S s

Treat itinerant electrons perturbatively: introduce effective bosons for an interaction between itinerant electrons and integrate out itinerant fermionic degrees of freedom

1 2 3 1 1 2 2 3 3

1 2 3 1 2 3

1loc

,

,,

[ ( 2 )( 2 )]

( 2 ) ( 2 ) (

) .

2 )

..( 2i i

m m mm m mq q q q q q qmm

q mm

mm m m m m mq q q abcd q q a q q b q q c

q m

mq q q q q q

L L R I I

I I I

I

-- - -

- - - - - -

= -

t t t

t t t

t

t S S

S S S

S

“bare” quadratic term

quartic interaction

(similar to s-d Shubin-Vonsovskii model).

it loc

The dynamic susceptibility

,it

,loc

10 1 0 2 0,it it loc

0 1 2 2 (4) 0,loc it

( )

( ) 4 22 ( ) 4 4

q q

q q

q q

q q q

R I R RR

I II I I

--

-

- = - -

RKKY

“Moriya”correction

Influence of itinerantelectrons on local momentdegrees of freedom

bare

bare

Exchange integrals and magnetic properties can be extracted

Return to a-iron

Return to a-iron

How do we recover RKKY exchange for a-iron?Assume: 𝜒 irr (𝐪 )❑=1/ 𝐼+𝜒 ′

irr (𝐪) 𝜒 ′irr ≪1/ 𝐼❑2

I ~ 1 eV – extracted in this way, in agreement with performedanalysis and band structure calculations

Size of local moment

Orbitally-resolved DOS

U = 4 eV, = 10 eV-1

LDA

a-Iron can be viewed as asystem in the vicinity of an orbital-selective Motttransition (OSMT)

Ratio of moments

The size of the instantaneous and effective moment

1( , , ) ( ) ( ) ( ) ( )4

( ) ( )

xc

xc

f fJ B

B

m m

m m

m

e e

-=

-

r r r r r r

r r

Requires a ‘reference magnetic state’ to calculate exchange integrals:

In which cases one can avoid use of the ‘reference state’ ?

Example: (one-band) Hubbard model at half filling due to metal-insulator transition the electrons are localized, Jij=4t2/U

Reference state is needed to introduce magnetic moment in an itinerant approach

r r'

(A. I. Liechtenstein, M. I. Katsnelson, et al.)

Magnetic exchange:L(S)DA formula:

NMFM

(2,0,0)

(0,0,)FM, bcc