Effects of electronic correlations in
iron and iron pnictides
A. 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)
a - bcc, g - fcc, e - hcp
ag
Ts =1185 K
a-iron: TC = 1043K, meff =3.13mB
g-iron: qCW =-3450K, meff =7.47mB
Arajs. J.Appl.Phys. 31, 986 (1960)
Parsons, Phil.Mag. 3, 1174 (1959)
• 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
a-Iron shows features of both, itinerant (fractional magnetic moment)
and localized (Curie-Weiss law with large Curie constant) systems
• 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)
 Local moments
(Heisenberg model)
 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?
Dynamical Mean Field Theory
Energy-dependent effective medium theory
 ( )
The self-energy of the embedded atom coincides
with that of the solid (lattice model), which is
approximated as a k-independent quantity
-1
( ) =  -1 ( ) - Gloc
( )
Gloc ( ) = 
k
1
 - e k  m - ( )
A. Georges et al., RMP 68, 13 (1996)
Spin-polarized LDA+DMFT
U = 2.3 eV, J = 0.9 eV
Magnetic moment 3.09
Critical temperature 1900 K
Lichtenstein, Katsnelson, Kotliar, PRL 87, 67205 (2001)
(3.13)
(1043K)
a (Bcc) iron: band structure
eg
A. Katanin et al.,
PRB 81, 045117
(2010)
t2g
t2g и eg states are
qualitatively different and
weakly hybridized
Correlations can “decide”,
which of them become local
a-iron: orbitally-resolved self-energy
Imaginary frequencies
t2g states - quasi-particles
eg states - non-quasiparticle!
A. Katanin et al., PRB 81, 045117 (2010)
Comparison to MIT:
Linear for the Fermi liquid
Divergent for an insulator
Bulla et al., PRB 64, 45103 (2001)
Self-energy and spectral functions
at the real frequency axis
Real frequencies
a-Fe
Comparison to MIT:
From: Bulla et al.,
PRB 64, 45103 (2001)
How to see local moments:
local spin correlation function
Local moments are stable
when
S(0)S()
S z ( ) S z (0)  const
 Fulfilled at the conventional
Mott transition. Can it be
fulfilled in the metallic
phase ?
J=0.9
J=0

A. Katanin et al., PRB 81, 045117 (2010)
Fourier transform of spin
correlation function
 ( ) =
meff2
3T
f ( / T )
Fourier transform of spin correlation
function
 ( ) =
meff2
3T
Local moments formed out of eg states do exist in iron!
f ( / T )
Which form of 𝜒𝑙𝑜𝑐 𝜔 one can expect for
the system with local moments?
 S (0) S ( )  const( )

 (in ) =  d  S z (0) S z ( ) ei    S 2  n ,0 / 3T
n
0
 Broaden delta-symbol:
 (in ) =
meff2
g
3T | n | g
 ( ) =
meff2
ig
3T   ig
g is the damping of local collective excitations
Re  ( ) =
Im  ( ) =
meff2
g2
3T   g
2
m
2
g
3T  2  g 2
2
eff
For a-iron:
𝜇eff = 3.3𝜇𝐵
𝛾 ≈ 𝑇/2
(𝜔 ≪ 𝐽)
Curie law for local susceptibility
eg
t2g
Total
local moment
 = g m B p( p  1) / (3T )
2
2
p(eg) = 0.56
p(t2g) = 0.45
p(total)=1.22
agrees with the
experimental data (known also after A.Liechtenstein,
M. Katsnelson, and G. Kotliar, PRL 2001)
Effective model
The local moments are coupled via RKKY-type of exchange:
J
H effd = H eg  H t2 g  H t2 g -eg  (U  - )  N i nim - 2 J  Si sim
2 i , ,mt2 g '
i , mt2 g '
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
TN≈100K
 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)?
LDA DOS
The peak in eg band is shifted by 0.5eV downwards with respect to
the Fermi level
g-(fcc) iron
P. A. Igoshev et al., PRB 88, 155120 (2013)
More itinerant than a-iron ?
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
QX=(0,0,2)
Comparison of energies in LDA approach
Shallcross et al., PRB 73, 104443 (2006)
SDW2
Magnetic state: Heisenberg model picture
Heisenberg model
A. N. Ignatenko, A.A. Katanin,
V.Yu.Irkhin, JETP Letters 87, 555
(2008)
For stability of (0,0,2) state one needs J1>0, J2<0.
The polarization bubble, low T
LDA
LDA+DMFT
m
k
k+q
T=290К
2(1,0,0)
2(1/2,1/2,1/2)
2(1,1/2,0)
m'
Experimental magnetic structure
q = (2/a) (1, 0.127, 0)
Tsunoda, J.Phys.: Cond.Matt. 1, 10427 (1989)
Naono and Tsunoda, J.Phys.: Cond.Matt. 16, 7723 (2004)
Fermi surface nesting
Colorcoding: red – eg, green – t2g, blue – s+p
(0,x,2) state is supported by the Fermi surface geometry –
an evidence for itinerant nature of magnetism
The polarization bubble, high T
LDA
LDA+DMFT
T=1290К
Uniform susceptibility
m
k
1/
k
From high-temperature part: 𝜇𝑒𝑓𝑓 = 4.07𝜇𝐵 (𝑝𝐶 ≅ 3/2)
m'
g-(fcc) iron
𝜇𝐶𝑊 = 7. 7𝜇𝐵
exp  -1500 K
 DMFT  -2700 K
TN  100 K (small particles)
 Strong frustration!
 Nonlocal correlations are important
The experimental value of the
Curie constant is reproduced
by the theory, although the
absolute value of paramagnetic
Curie temperature appears too
large
Magnetic exchange in g-iron
𝜒0
𝜒𝐪 =
1 − 𝐽𝐪 𝜒0
#
𝜒irr (𝐪)
𝜒𝐪 =
1 − Γ 𝜒irr (𝐪)
1
J z/2,K -669
2
3
4
5
6
173
-449
17
-25
-123 -116
−1
𝐽𝐪 = −[𝜒irr (𝐪)]
7
8
29
+𝐶
𝐽𝟎 = −2500𝐾
𝐽𝐐 = 1200𝐾
The Neel temperature is much larger than the experimental one,
similar to the result of the Stoner theory:
o Paramagnons
o Frustration, i.e. degeneracy of spin susceptibility in different directions
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
Electronic correlations
Damped qp
states
qp states
No qp states
387K
580К
1160К
xy
0.142
0.242
0.454
xz, yz
0.131
0.163
0.306
3z2-r2
0.054
0.092
0.228
x2-y2
0.053
0.101
0.334
dxz, dyz, dxy states can be
more localized
Local susceptibility
387K
580К
1160К
xy
-0.142
-0.242
-0.454
xz, yz
-0.131
-0.163
-0.306
3z2-r2
-0.054
-0.092
-0.228
x2-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 responsible
for the part of linear-dependent
term in (T)
S. L. Skornyakov, A. Katanin,
and V. I. Anisimov, PRL ’ 2011
Summary
 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 susceptibilities
proves to be helpful in classification of different substances regarding
the degree of local moment formation
In alfa-iron:


The existence of local moments is observed within the
LDA+DMFT approach
The formation of local moments is governed by Hund
interaction
In gamma-iron:


Local moments are formed at high T>1000K, where this
substance exist in nature, but not at low-T (in contrast to alfairon); the low-temperature magnetism appears to be more
itinerant
Antiferromagnetism is provided by nesting of the Fermi surface
Conclusions
In the iron pnictide:



Electronic correlations are important, but, similarly to g-iron, local
moments may be formed at large T only
Different orbitals give diverse contribution to magnetic
properties
Linear behavior of uniform susceptibility is (at least partly) due to peaks
of density of states near the Fermi level
Thank you for attention !
Spin correlation functions
Spectral functions
Damped qp
states
qp states
No qp states
Effective model and diagram
technique
H effd = H t2 g  H eg - 2 I
 Ss
i im
i ,mt2 g
I
 (U  - )
nim nim 

2 i , ,meg ,mt2 g
(similar to s-d Shubin-Vonsovskii model).
Treat eg electrons within DMFT and t2g electrons perturbatively
Simplest way is to decouple an interaction and integrate out
t2g electrons
 [R
L = Leg 

-1 m m
mm q - q
t t
-
mm
q
( t  2 ISq )( t
m
q
m
-q
 2 IS- q )] 
q ,mm
“bare” quadratic term
mm
m
m
m
   mm
(
t

2
I
S
)
(
t

2
I
S
)
(
t
q1q2 q3 ,abcd
q1
q1 a
q2
q2 b
q3  2 IS q3 ) c





qi ,mi

( t m
 2 IS
)
 ...
Diagram technique: perspective

mm
q


=

mm
 mm
q1q2 q3 ,abcd =
 q0,e =
g
abcd
a b c d
(4),
=

S
q1q2q3
q1 Sq2 Sq3 S- q1 -q2 -q3  c ,eg =
The dynamic susceptibility
 ( R  q ,t2 g  I ) R  q R 





R

q
q ,eg 

“Moriya”
correction
bare
 (  q0,t2 g )-1     t02 g  4 I 2    e0g
=

-2 I  q



(  q0,eg )-1 - 4 I 2  q  4 I 2 (4)  t02 g 
bare
RKKY
-2 I  q
-1
Influence of itinerant
electrons on local moment
degrees of freedom
• Two different approaches to magnetism of transition metals
(and explaining Curie-Weiss behavior):
- Itinerant (Stoner, Moriya, …)
- Local moment (Heisenberg, …)
Local moments in transition metals
Since they are (good) metals, at first glance no ‘true’
local moments are formed
However, under some conditions the formation
of (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
Can one unify these approaches
(one band: Moriya, degenerate bands:
Local moments in transition metals
Since they are (good) metals, at first glance no ‘true’
local moments are formed
However, under some conditions the formation
of (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
Dependence on
imaginary frequency
Paramagnetic LDA+DMFT
U = 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.
 k0,i
“paramagnon”
1 - U  k0,i  
n
n
0abab
0

k ,i
4/3
  T  0abab 

T
0
b =1...3
k ,i 1 - U  k ,i  
n
n
n
Curie-Weiss-like susceptibility
Frustration in Heisenberg FCC model
Polarization bubble
m
m'
eg
t2g-e2g
t2g
G. Stollhoff, 2007
Diagram technique: perspective

mm
q


=

mm
 mm
q1q2 q3 ,abcd =
 q0,e =
g
abcd
a b c d
(4),
=

S
q1q2q3
q1 Sq2 Sq3 S- q1 -q2 -q3  c ,eg =
S(0)S()
Spin correlation function at different U
eg
almost flat !
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.
 k0,i
“paramagnon”
1 - U  k0,i  
n
n
0abab
0

k ,i
4/3
  T  0abab 

T
0
b =1...3
k ,i 1 - U  k ,i  
n
n
n
Curie-Weiss-like susceptibility
Effective model
H effd = H t2 g  H eg - 2 I
it
loc

i , mt2 g
Sisim
I

 (U - )  N i nim
2 i , ,mt2 g
(similar to s-d Shubin-Vonsovskii model).
Treat itinerant electrons perturbatively:
introduce effective bosons for an interaction between itinerant electrons and
integrate out itinerant fermionic degrees of freedom
L = Lloc 
m
m
 [ Rmm-1  t mq t m- q -  mm
q (t q  2 IS q )(t - q  2 IS - q )] 



q , mm
“bare” quadratic term
 
mm m m
q1q2 q3 , abcd
m
q2
(t  2 IS q1 ) a (t  2 IS q2 )b (t
m
q1
m
q3
 2 IS q3 )c
qi , mi

(t m- q1 -q2 -q3  2 IS - q1 -q2 -q3 )
quartic interaction
 ...
The dynamic susceptibility
 ( R  q ,it  I ) R

q R

q R 
 q ,loc 
“Moriya”
correction
bare
0
 (  q0,it ) -1     it0  4 I 2    loc
=

-2 I  q


-2 I  q

(  q0,loc ) -1 - 4 I 2  q  4 I 2  (4)   it0 
bare
-1
RKKY
Influence of itinerant
electrons on local moment
degrees of freedom
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:
𝜒irr (𝐪) = 1/𝐼 + 𝜒 ′ irr (𝐪)
−1
𝐽𝐪 = −(𝜒irr )
𝜒′irr ≪ 1/𝐼
+ 𝐶 = 𝐶 ′ + 𝐼 2𝜒 ′ irr (𝐪)
I ~ 1 eV – extracted in this way, in agreement with performed
analysis and band structure calculations
Size of local moment
Orbitally-resolved DOS
LDA
U = 4 eV,  = 10 eV-1
a-Iron can be viewed as a
system in the vicinity of
an orbital-selective Mott
transition (OSMT)
Ratio of moments
The size of the instantaneous and
effective moment
Magnetic exchange:
r
L(S)DA formula:
J (r , r  ,  ) =
f m  - f 
1
 m  (r ) Bxc (r )   (r )  (r  )

4 m  - e m   e 
 Bxc (r  ) m  (r  )
(A. I. Liechtenstein, M. I. Katsnelson, et al.)
Requires a ‘reference magnetic state’ to calculate exchange
integrals:
Reference state is needed to introduce magnetic
moment in an itinerant approach
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
r'
(2,0,0)
NM
FM
(0,0,)
FM, bcc