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
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
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
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+-)
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
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
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
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
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