Interplay between magnetism and superconductivity in Fe-

pnictides

Institute for Physics Problems, Moscow, June 30, 2010

Andrey Chubukov

University of Wisconsin

IRON AGE (1200 - 550 B.C.E.)

RFeAsO (1111)R = La, Nd, Sm, Pr, Gd

AFe2As2 (122)A = Ba, Sr, Ca

LiFeAs (111)

Fe-Pnictides:

Binary componds of pnictogens A pnictogen – an element from the nitrogen group N,P, As,Sb,Bi

LaOFeP Fe(SeTe)

Tc,max =57K (SmFeAsO)

Hideo Hosono, TITech

Fe-pnictides: Fe-pnictides: May 2006

2006

43K Tc ,FO SmFeAs

26K Tc ,FLaFeAsO

xx-1

xx-1

2008

LaFeAsO1-xFx

La1-xSrxFeAsO

SmFeAsO1-xFx

CeFeAsO1-xFx

PrFeAsO1-xFx

NdFeAsO1-xFx

GdFeAsO1-xFx

SmFeAsO1-xFx

SmFeAsO1-x

GdFeAsO1-x

Gd1-xThxFeAsO

DyFeAsO1-xFx

TbFeAsO1-xFx

Tb1-xThxFeAsO

Ba1-xKxFe2As2

Sr1-xKxFe2As2

Eu1-xLaxFe2As2

Ca1-xNaxFe2As2

Eu1-xKxFe2As2

Li1-xFeAs

-FeSe1-x

BaNi2P2

LaO1-xNiBiLaOFePLaO1-xFxFeP

LaONiP

-FeSe

SrNi2As2

BaCoxFe2-xAs2

SrCoxFe2-xAs2

BaNixFe2-xAs2

FeSe0.5Te0.5

2008

Tc (K)

courtesy of J. Hoffman

Phase diagram: magnetism and superconductivity

BaFeBaFe22(As(As1-x1-xPPxx))22

Fernandes et al

Matsuda et al

Luetkens et al

Crystal structure

2D Fe-As layers with As above and below a square lattice formed by Fe

LaFeAsO

Folded BZ

Unfolded BZ

Pnictides Cuprates

Are pnictides similar to cuprates?

Parent compounds are antiferromagnets

Superconductivity emerges upon doping

Pnictides Cuprates

Are pnictides similar to cuprates?

metal Mott insulator/Heisenberg AFM

Metallic behavior of parent compoundsMetallic behavior of parent compounds

TN

Resistivity in the cuprates

parent compounds(magnetic)

Band theory calculations agree with experiments Band theory calculations agree with experiments Lebegue, Mazin et al, Singh & Du, Cvetkovic & Tesanovic…

2 circular hole pockets around (0,0)

2 elliptical electron pockets around ) (folded BZ), or and (unfolded BZ)

Electron Fermi surfaceHole Fermi

surface

NdFeAs(O1-xFx) (x=0.1)

A. Kaminski et al.

Hole pockets near (0,0) Electron pockets near ()

dHVaARPES

LaFeOPA. Coldea et al,

Ba06K04Fe2As2

H. Ding et al.

A. Kordyuk et al

LiFeAs

Fe-pnictides are doped metals

High Tc cuprates are doped Mott insulators

A simple way to understand the difference between cuprates and pnictides

4s[Ar]3d :Cu ,4s3d [Ar] :Fe 11026

Cuprates: 3d Cu 92

Tesanovic, Physics 2, 60 (2009)

one hole in a filled d-shell(1 “free” fermion per cite: half-filling)

U

Only when doped with holes (or electrons) do cuprates turn into superconductors

Cu ,Fe 22 9

Pnictides: 3d Fe 62

4 holes per cite – multiband structure. chemical potential lies in the gap

LaOFeAs

Itinerant approach

Interacting fermions with hole and electron Fermi surfaces, no localization of electronic states

What are generic, model-independent features of Fe-pnictides?

I will skip magnetism and focus only on

superconductivity

Magnetic and superconducting properties are both interesting

Superconductivity

Electron-phonon interaction is probably too weakElectron-phonon interaction is probably too weak

Boeri, Dolgov, & Golubov, Singh & Du

=0.44 for Al

Too small to account for Tc ~ 50K

I. Mazin et al., PRL 101, 057003 (2008)

How about using the “analogy” with the cuprates andassume that the pairing is mediated by (spin) fluctuations

Pairing due to el-el interaction

Cuprates

)k cos - k (cos )( yx0

0

00 - ),( , )0( kk

Mazin et al, Kuroki et al…. peaked at ()

Pnictides

sign-changing s-wave gap (s+-)

Some experimentsare consistent withthe sign-changing, no-nodal s+- gap

Almost angle-independent gap(consistent with no-nodal s-wave)

NdFeAsO1-xFx

T. Kondo et al., arXiv:0807.0815

1a. Photoemission in 1111 and 122 FeAs 1a. Photoemission in 1111 and 122 FeAs

D. Evtushinsky et al

Ba1-xKxFe2As2

1b. Penetration depth behavior in 1111 and 122 FeAs 1b. Penetration depth behavior in 1111 and 122 FeAs

Carrington et al

“Exponential” behavior at low T(or, at least, a very flat behavior)

Y. Matsuda

SmOFeAsSmOFeAs Ba1-xKxFe2As2

Ba0.46K0.56Fe2As2

1c. Neutron scattering – resonance peak below 2D 1c. Neutron scattering – resonance peak below 2D

D. Inosov et al.

kk - needs one

:say Theorists

Eremin &KorshunovScalapino & Maier

The “plus-minus” gap is the best candidate

s+- gap

Other experiments,however, indicate that the gap may have nodes

2a. The behavior of LaOFeP, Tc =5K 2a. The behavior of LaOFeP, Tc =5K

Carrington et al

Linear in T dependence at small T

Penetration depth Thermal conductivity

Matsuda et al

Residual term in /T,H1/2 field dependence

Consistent with line nodes

2b. The behavior of BaFe2b. The behavior of BaFe22(As(As1-x1-xPPxx))22, Tc =30K , Tc =30K

Y. Matsuda et al

(BaK)FeAs

BaFe(AsP)

Consistent with line nodes

2c. Residual term in c-axis thermal conductivity of 2c. Residual term in c-axis thermal conductivity of overdoped Co-Ba122overdoped Co-Ba122

c-axis conductivity a-axis conductivity

J-Ph. Reid et al

Back to a simple reasoning

Problem: how to get rid ofan intra-band Coulomb repulsion?

Cuprates Pnictides

FS

0 d )(

Coulomb repulsion cancels out, only d-wave, interaction matters

FS

0 d )(

Intra-band repulsion does notcancel and has to be overtaken by a interaction

hole FS

u4

u3

electron FS

Intra-band repulsion u4

Pair hopping

interaction

u4

,u - u )( 43SCs

If the intra-band repulsion (u4) is stronger than the pair hopping (u3), the pairing interaction is repulsive

TE

log )( 1

1

FSC

SC

s

s s+- superconductivity, a simple RPA

Pairing interactions: A toy model: one hole and one electron FSs.

How to overcome intra-pocket Coulomb repulsion is the most essential part of the theory of itinerant superconductivity in Fe-pnictides

Electronic Structure: a zoo of p- and d-states bandsElectronic Structure: a zoo of p- and d-states bands

L. Boeri, O.V. Dolgov, and A.A. Golubov, PRL 101, 026403 (2008)

Strong hybridization of all Fe-orbitals

10 Fe d-bands in thewindow of +- 2 eV

12 O and As p-bandsfrom -6 eV to -2 eV

La f-bands at around -3 eV

I. Pnictides are multi-orbital systems

What multi-orbital/multi-band structure means for our story?

• Interactions between low-energy fermions are dressed by orbital coherence factors and in general have strong angular dependence.

Effective intra-band interaction:

2 cos u~ up)(k, 33 Effective inter-band interaction:

4u p)(p, k)(k,

(previously we only had u3)

Exercise in trigonometry with s-wave interactions:

This is still s-wave!

Solve three coupled BCS equations for the gaps:

When is non-zero, the solution exists for arbitrary u3/u4

and for arbitrary sign and magnitude of ,

When is zero, the solution for Tc exists only for u3>u4

but the gap along electron FS has nodes for u3>u4.

1

s+- gap with the nodes on electron FS

s+- gapwith no nodes

d-wave gap with the nodes on both FS

Coulomb repulsion is the strongest – the s+- gap withthe nodes along electron FS is most likely scenario

Superconducting phase diagram

How can the system develop a no-nodal gap (still of s+- symmetry) ?

SmOFeAsSmOFeAs

RG helps:

u4 and u3 are bare interactions,at energies of a bandwidth

For SC we need interactions at energies smaller than the Fermi energy

E

EF ~ 0.1 eV W ~3-4 eV| |

0

Couplings flow due to renormalizations by particle-particle and particle-hole bubbles

We know this story for conventional (phonon) superconductors

EF ~ WD

0 ECoulomb repulsion is renormalized downby the renormalization in the particle-particlechannel (McMillan-Tolmachev renormalization)

If only Cooper channel is involved, this cannot change a repulsion into an attraction

In our case, there are renormalizations in both particle-particleAND particle hole channel. This implies that we need to construct parquet RG to analyze the system flow between W and EF

(H. Shultz, Dzyaloshinskii & Yakovenko)

W/Elog )u(u 1

)u(u u u

034

03434

One-loop parquet RG One-loop parquet RG

u3 is pushed up by u1 , which gives rise to SDW

Intra-band repulsion u4

Pair hopping

interaction

Inter-band density-densityand exchange interaction

SDW

u0L

If the tendency towards SDW is strong, u3 becomes larger than u4, leading to an s+- superconductivity with no-nodal gap

1

s+- gap with the nodes on electron FS

s+- gapwith no nodes

d-wave gap with the nodes on both FS

If RG renormalization is strong, Coulomb interactionis overshadowed, and the s+- gap may have no nodes along the electron FS

Superconducting phase diagram

Less SDW fluctuations

Underdoped 1111, 122 FeAs LaFeOP, Overdoped FeAs

The role of spin fluctuations

SC

sf

)u (u u 231

sf

Spin-fluctuation exchange is an additional dynamical interaction, which acts in parallel with the pair hopping.

The real issue is what sets the upper cutoff for the pairing: the scale at which u3 =u4, or the dynamics of the spin-fluctuatioon exchange

Interplay between magnetism and superconductivity

We can re-write these as equations for SDW, CDW, and SC vertices

Interactions in all channels grow

- -

One-loop RG Flow – all channelsOne-loop RG Flow – all channelsSDW with real order parameter

Extended s-wave

CDW with imaginary order parameter

Above EF

Lower boundary for parquet RG is the Fermi energy, EF

O(6) fixed point:3 for SDW, 2 for SC,1 for CDW

Below EF – decoupling between SDW and SC channels

R4

R3SC

R3

R1SDW

RiFi

u u ,u u

u )E ~ (E u :aluesBoundary v

For a perfect nesting, whichever vertex is larger at EF, wins

Perfect nesting – SDW wins

Non-perfect nesting –SDW vertex remains the strongest, but the SDW instability is cut, and s+- SC wins

Either first-order transition between SDW and SC states, or co-existence, depending on parameters

dopingdoping

depending on thedegree of ellipticity of electron FSs

Fernandes, Schmalian, et al,Vorontsov, Vavilov, AC

How important is the fact that there are 5 bands and the orbital structure changes along the FS?

Coldea et al, dHvA

Mazin & Schmalian Graser, Maier, Hirschfeld, ScalapinoThomale, Platt, Hanke, Bernevig.D-H Lee, Vishvanath, Honerkamp, Benfatto, Grilli, Kuroki, Imada…At least 2 hole and

two electron FSs

additional 3D FSs

Detailed studies:

These extra features give rise to some qualitative changes, but do not change the overall picture of s+- pairing

IIa. RG for 2 hole and 2 electron FSs

more parameters, more equations

SC

SC SDW

SDW

1hole 1 electron FSs 2 hole 2 electron FSs

Still, SC vertex changes sign under RG

Thomale et alMaiti, AC

Summary: the phase diagram:

1

s+- gap with the nodes on electron FS

s+- gapwith no nodes

Less SDW fluctuations

Underdoped 1111, 122 FeAs LaFeOP, Overdoped FeAs

Coulombrepulsion

Angle dependence of the interaction

Position depends on interactions, FS geometry, etc…

Conclusions:

Superconductivity is the result of the interplaybetween intra-pocket repulsion and the pair hopping.

If the tendency towards SDW is strong, pair hopping increases in the RG flow, and the system develops an s+- gap without nodes, once SDW order is eliminated by doping.

If the tendency towards SDW is weaker, intra-pocket repulsion remains the strongest. The system still becomes an s+- supercnductor, but the gap has nodes along the two electron Fermi surfaces.

Fe-pnictides are itinerant systems, no evidence for Mott physics

THANK YOU

IIb. Does the solution for s+- gap with nodes exists for 2 hole and 2 electron FSs when the Coulomb repulsion is the strongest (i.e., when RG does not work)?

more parameters, more equations

2 hole 2 electron FSs, hence two hole

two electron

Still, the solution for s+- gap with nodes along electron FS does exist!

The equation on L= log Tc becomes 4th order

1,2

Matano et al

Knight shift NMR relaxation rate

Non-exponential behavior!

1. NMR and Knight shift in 1111 FeAs1. NMR and Knight shift in 1111 FeAs

3. Penetration depth behavior in 1111 and 122 FeAs 3. Penetration depth behavior in 1111 and 122 FeAs

Carrington et al

“Exponential” behavior at low T(or, at least, a very flat behavior)

Y. Matsuda

SmOFeAsSmOFeAs Ba1-xKxFe2As2

Ba0.46K0.56Fe2As2

A “derivation”: take a toy model of two electron orbits + hybridization + Hubbard-typerepulsive interactions between fermions from thesame orbital (U11) and from different orbitals (U12)

+

1p2

02

1202

11F

32

02

1202

11F

145

-

2 tan

sin 2U ) cos (1 U 2

u u

sin 2U ) cos (1 U 2

u u u

p

pp

pp

pp

The couplings depend on the energies of the two orbitals, and on hybridization

All coupling are positive (repulsive), andintra-band repulsion is stronger than the inter-band pair hopping

How we proceed in the cuprates:

,...3k cos 3k cos ,2k cos 2k cos ,k cos k cos yxyxyxB1g

nodalantinodal

)k cos k (cos 2

yxB1g

Abanov et al 09

actual gap

)2(cosor

k cos k cos yx

... 6 cos B 2 cosA 1gB

) 2 cos ( 1gB