Eliashberg Theory

G.A. Ummarino LATEST group

Institute of Physics and Materials Engineering DISAT

Applied Science and Technology Departement Politecnico di Torino

September 2013

)()(V '', kDkDkkV εωθεωθ −−−≈

( ) ( )( )

( )∑< Δ+

Δ+

Δ−=Δ

Dk

k

k

kT

T

T

TVNTωε ε

ε

'

'

22'

22'

2tanh)0(

Two input parameters: the representative energy of bosonic spectra ωD and the electron-boson coupling N(0)V. (ωD can be obtained by experimental data )

Output: Δ(T) and Tc

BCS THEORY

])0(/1exp[2)0( VNT D −==Δ ω])0(/1exp[13.1 VNT Dc −= ω

53.3/2 =Δ cT

The theory is approximatevily correct

only in weak coupling λ≈N(0)V<0.6

for λ > 0.6 the theory is correct

only in a qualitative way and not always

43.1/)( =Δ cc TTC γ

λ Tc (K) 2Δ/kBTc ΔC(Tc)/γTc

Al 0.43 1.18 3.535 1.43In 0.81 3.40 3.791 1.79Pb 1.55 7.22 4.497 2.77Nb3Sn 1.7 17.90 4.567 2.64Bi 2.45 6.11 4.916 2.03Pb0.75Bi0.25 2. 76 6.91 5.119 2.27

Range of application of the BCS theory

Eliashberg theory

λωD/EF <<1, first order Feynman diagrams

Input parameters of Eliashberg theory

Bosonic spectral function α2(ω)F(ω)

Coulomb pseudopotential µ*(ωc)

∫∞+

ω

ωαω=λ

0

2 )(2 Fd ⎥⎦

⎤⎢⎣

⎡= ∫

∞+

0

2

log)()log(2exp

ωωα

ωωλ

ωFd

Existence of Fermi surface

λλ

Nambu-Gorkov formalism

Dyson equation

λωD/EF <<1

only first order Feyman diagrams

Now we impose self-consistency by replacing the esplicit expression

for the Green functions inside the self energy expression.

Iterative solution

Determination of Tc

Δ=Δ(iωn=0) is the value of the superconductive gap only in weak coupling regime:λ<<1

Padè approximants T<Tc/10

Three different but equivalent formulations: imaginary, real and mixed

Lead

If there are magnetic impurities this method don’t work (of course always T<<Tc)

Input parameters: α2F(ω) and µ*

Numerical solution of Eliashberg Equations

Complex function Δ(ω,T) and Z (ω,T)

All superconductive physical observables

are simple functions of Δ(ω,T) and Z(ω,T)

in BCS theory Δ(ω,T) is a number

and Z (ω,T)=1

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

1.2

α2 F(ω

)

ω (meV)

0 5 10 15 20 25 30 35 40 45-3

-2

-1

0

1

2

3

4

Pb T=0 KTc=7.22 K, Δ=1.4 meV,

λ=1.55, µ*(ωc)=0.131 Re(Δ(ω)) Im(Δ(ω))

Re(Δ(ω)),

Im(Δ(ω))

(meV

)

ω (meV)

0 5 10 15 20 25 30 35 40 450.0

0.5

1.0

1.5

2.0

2.5

3.0

Pb T=0 KTc=7.22 K, Δ=1.4 meV,

λ=1.55, µ*(ωc)=0.131 Re(Ζ(ω)) Im(Ζ(ω))

Re(Ζ(ω)),

Im(Ζ(ω))

ω (meV)

BCS limit

at very low T the current derivative is proportional at the

superconductive density of states

Tunneling: junction S I N

( ) ωω =Δ ],Re[ T the superconductive gap

Real axis Eliashberg Equations at T=0

( ) ( ) ( ) ( )[ ] )'(','' *

0 1 ωωθµωωωωωω −−⋅⋅=Ζ⋅Δ +

+∞

∫ cKnd

( )( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δ−=

'''Re'222ωω

ωωn

( )[ ] ( ) ( )',''10 2 ωωωωωω −

+∞⋅⋅=⋅Ζ− ∫ Knd

( ) ( ) ( ) ⎥⎦

⎤⎢⎣

⎡−Ω+−

±+Ω++

⋅ΩΩ⋅Ω= ∫∞+

± δωωδωωαωω

iiFdK

'1

'1',

0

2

( ) ( )( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

Δ−

Δ=

'''Re'221ωω

ωωn

( )exp

]Re[Δ=Δ

=Δ

c

ωω*µ

In the past α2F(ω) and µ*

had obtained by experimental data

(also for no superconducting materials

via the proximity effect)

or was free parameters

but now can be calculated by

DENSITY FUNCTIONAL THEORY (DFT)

No free parameters !

Physical observables can be calculated:

Critical temperature

Gap value as function of temperature

Superconductive density of states

Upper critical field as function of temperature

Specific heat as function of temperature

NMR

Magnetic susceptivity

Isotopic effect

Etc…

Tc Equations

BCS λ<<1 analityc solution

Eliashberg λ<1.5 µ*<0.15 num solution

Eliashberg λ>10 analityc solution

⎥⎦

⎤⎢⎣

⎡

−

+−= *

1exp13.1µλλωDCT

⎥⎦

⎤⎢⎣

⎡

+−

+−=

)62.01()1(04.1exp

2.1 * λµλλωD

CT

λωDCT 183.0≈

Isotropic order parameter Δ(ω) HTCS

Conduction bandwidth infinite Fullerenes, HTCS

Half filling of conduction band HTCS

Normal density of state constant around the Fermi level Nb3Sn, PuCoGa5

One conduction band MgB2, iron pnictides

Validity of Migdal Theorem λωD/EF<<1, HTCS, fullerenes Spin of Cooper pair =0 Sr2RuO4

No effect of disorder and magnetic impurities all superconductors

Standard Eliashberg Equations

three different but equivalent formulations:

real axis, imaginary axis and mixed

Approximations of Eliashberg Theory, in red the possible generalizations

Breakdown of Migdal theorem corrections at the first order in λωD/EF at T=Tc

L. Pietronero, S. Str¨assler and C. Grimaldi, Phys. Rev. B 52, 10516 (1995)

C. Grimaldi, L. Pietronero and S. Str¨assler, Phys. Rev. B 52, 10530 (1995)

A λeff >> λ in the standard Eliashberg equations can reproduce the vertex corrections

Stable system

O.V. Danylenko, O.V. Dolgov, PRB, VOLUME 63, 094506, (2001)

O.V. Dolgov and V.V. Losyakov, Phys. Lett. A 190, 189 (1994)

λ>>1 ? YES

(λz>0)

COPPER OXIDES (overdoped), PuCoGa5 etc (d-wave)

F. Jutier, G.A. Ummarino, J.C. Griveau, F. Wastin, E. Colineau, J. Rebizant, N. Magnani and R. Caciuffo, Phys. Rev. B 77, 024521 (2008); D. Daghero, M. Tortello, G.A. Ummarino, J.-C. Griveau, E. Colineau, A.B. Shick, R. Caciuffo, J. Kolorenc and A.I. Lichtenstein, Nature Communications 3, 786, (2012)

PuCoGa5 Tc=18 K

Magnesium diboride MgB2 Tc=39.4 K (2001)

Two isotropic order parameters

Two renormalization functions

Four spectral functions

Two conduction bands: π and σ

Four Coulomb pseudopotential

Eliashberg Equations

0 10 20 30 40 50 60 70 80 90 100 1100.0

0.5

1.0

1.5

2.0

α2 F(Ω

)

Ω (meV)

α2Fππ

(Ω), λππ

=0.448 α2F

σσ(Ω), λ

σσ=1.017

α2Fσπ

(Ω), λσπ

=0.213 α2F

πσ(Ω), λ

πσ=0.155

All physical observables are in agreement with theoretical predictions!

No free parameters!

Proximity effect

H. G. Zarate and J. P. Carbotte, PRB 35, 3256 (1987)

The superconductivity in Fe-based layered compound is unconventional and mediated by antiferromagnetic spin fluctuations. This resulting state is an example of extended s-wave pairing with a sign reversal of the order parameter between different Fermi surface sheets.

3 bands with s-wave gap The phases of two of the three gaps are opposite Intraband coupling provided by phonons (P) Interband coupling provided by AFM spin fluctuations (SF)

ijSFiiSFiiP

ijPiiP

λλ

λ

λλ

<<

<

≈>>

35.0

0 Phonons mainly provide intraband coupling

The total e-ph coupling constant is very small

The SF mainly provide the interband coupling

Superconductivity in Fe-based compounds

Ba0.6K0.4Fe2As2: H. Ding et al, Europhys. Lett. 83 (2008)

[ ]

[ ]

∑

∑ ∑

∑

∑ ∑

Γ+Γ+

+−Λ+−Λ+

=

Γ−Γ+

+−−−Λ−−Λ

=Δ

∞+

−∞=

Δ

∞+

−∞=Δ

j nj

ZMij

Nij

jm

mj

ZmnijS

mnijP

n

nin

j njM

ijNij

jm

mcmj

cijmnijS

mnijP

nini

iNT

iNiiiiT

iZ

iNT

iNiiiiT

iZi

)()(

)()()(

)(

)()(

)()()()()(

)()(

*

ωπ

ωωωωωπω

ωω

ωπ

ωωθωωµωωωωπ

ωω

Multi-band s-wave imaginary- axis Eliashberg equations with phonons (P) and spin fluctuations (S)

Three-band s± Eliashberg equations

i,j =1-3 band index

( )

( )

∫

∫

∫

∫

∞+

∞+

∞+

∞+

Δ

Ω

ΩΩ=

Ω

ΩΩΩ=

−+Ω

ΩΩ=−Λ

−+Ω

ΩΩΩ=−Λ

Δ+

Δ=

Δ+=

0

0

2

0 22

0 22

2

2222

2222

)(2

)()(2

)()(

2

)()()(

2

)()()(

)()(

,)()()(

)()(

ijijS

ijijP

mn

ijmnij

S

mn

ijmnij

P

mjmjmjm

mjm

j

mjmjmjm

mjmm

jZ

Pd

Fd

iiP

dii

iiF

dii

iiZiZ

iiN

iiZiZ

iZiN

λ

αλ

ωωωω

ωω

αωω

ωωωω

ωω

ωωωω

ωωω

Electron-phonon coupling constant

Electron-spin fluctuation coupling constant

Three-band s± Eliashberg equations

)(2 ΩijFα

ij*µ

cω

)(ΩijP

Electron-phonon spectral functions (9)

Antiferromagnetic spin fluctuactions spectral functions (9)

Coulomb pseudopotential matrix elements (9)

electronic cut-off energy

ijMN ,Γ non-magnetic (N) and magnetic (M) impurity

scattering rates (9+9)

)0(iN

Inputs of the three-band Eliashberg equations

normal density of states at the Fermi level of the i-th band (3)

1)  λiisf = 0 (no intraband SF coupling)

2)  µ*ij=µ*ii =0 (no Coulomb pseudopotential)

3)  Γij=0 (no impurities) 4)  λii

ph = λijph = 0 (no phonon coupling)

To reduce the number of free parameters we exploit the fact that

Assumptions of the three-band Eliashberg model

Assumptions:

λiiph >> λij

ph ≈0 phonons mainly provide intraband coupling λij

sf >> λiisf ≈0 spin fluctuations mainly provide interband coupling

N1(0)/N2(0) N1(0)/N3(0) N2(0)/N3(0) BaFeCoAs 1.12 4.50

LaFeAsOF 0.91 0.53

BaKFeAs 1.00 2.00

SmFeAsOF 2.50 2.00

5)   αij2F(Ω)=Cij{1/[(Ω- Ωij)2+ Υ2

ij]-1/[(Ω+Ωij)2 +Υ2ij]}

6)  Ωij= Ω0=2Tc/5 and Υij=Ω0/2 (empirical law) (The peak in the spectrum is the same for all i,j)

7) Ni(0) from band theory calculations

Only two free parameters: λ31 , λ32 or λ21 , λ31

0 λ21ν21 λ31ν31 λ21 0 λ32ν32 λ31 λ32 0

where ν31=N3(0)/N1(0), ν32=N3(0)/N2(0) ν21=N2(0)/N1(0)

Assumptions of the three-band Eliashberg model

The coupling matrix λij is thus:

0 5 10 15 20 25 300.00.10.20.30.40.50.60.70.8

α2F(Ω

)

Ω (meV)

λij=

Tcexp(K) Tcth(K)

22.6 23.7 λ=2.8

28.6 26.5 λ=2.4

37.0 38.3 λ=2.8

52.0 52.8 λ=2.4

0

2

4

6

8

0,0 0,2 0,4 0,6 0,8 1,00

4

8

12

16

20

0,0 0,2 0,4 0,6 0,8 1,00

2

4

6

80

2

4

6

8

10

12

14

|Δ

i | (

meV

)

Ba(Fe0.9Co0.1)2As2

|Δi| (

meV

)SmFeAsO0.8F0.2

T/Tc

|Δi |

(m

eV

)

LaFeAsO0.9F0.1

T/Tcc

|Δi| (

meV

)

Ba0.6K0.4Fe2As2

G.A. Ummarino, Phys. Rev. B 83, 092508 (2011)

Imaginary-axis Eliashberg equations for the upper critical field

Three free parameters: the Fermi velocities that can be calculated by DFT but if they haven’t been yet calculated

Free electron model:

so that

Only a free parameter:

Andreev reflection: mechanism of superconductivity in the iron pnictides

Stuctures present at

D. Daghero, M. Tortello, G.A. Ummarino and R.S. Gonnelli, Rep. Prog. Phys. 74, 124509 (2011)

We put the solution of Eliashberg equations

in the Andreev reflection equation of conductance:

bosonic structures appear!

M. Tortello, D. Daghero, G. A. Ummarino, V. A. Stepanov, J. Jiang, J. D. Weiss,

E. E. Hellstrom, and R. S. Gonnelli1, PRL 105, 237002 (2010)

LiFeAs Tc=18 K

G A Ummarino, Sara Galasso and A Sanna J. Phys.: Condens. Matter 25 (2013) 205701

λtot,tr=0.77, λtot,sup=2 as HTCS

The same happens for BaFeCoAs, BaFeKAs…

All old superconductors

BKBO

fullerenes

Eliashberg theory works for:

MgB2

PuCoGa5 and,may be, for heavy

fermions and HTCS optimally doped

It does’t work (now, in this form)

for HTCS underdoped

HTCS overdoped iron pnictides

ESSENTIAL (not exaustive!) BIBLIOGRAPHY OF ELIASHBERG THEORY

1) D.J. Scalapino, in: R.D. Parks (Ed.), Superconductivity, Marcel Dekker Inc., NY, 1969, p. 449; 2) J.P. Carbotte, Rev. Mod. Phys. 62 (1990) 1028; 3) P.B. Allen, B. Mitrovich, Theory of superconducting Tc, in: H. Ehrenreich, F.

Seitz, D. Turnbull (Eds.), Solid State Physics, vol. 37, Academic Press, New York, 1982;

4) F. Marsiglio, J.P. Carbotte, in: K.H. Bennemann, J.B. Ketterson (Ed.), The Physics of Conventional and Unconventional Superconductors, Springer-Verlag;

5)A.V. Chubukov, D. Pines, and J. Schmalian: A Spin Fluctuation Model for d-wave Superconductivity in K.H. Bennemann and John B. Ketterson (eds.): Superconductivity (Springer, 2008) 6) D. Manske, I. Eremin, and K.H. Bennemann: Electronic Theory for Superconductivity in High-Tc Cuprates and Sr2RuO4 in K.H. Bennemann and John B. Ketterson (eds.): Superconductivity (Springer, 2008) 7) M.L. Kulic, Physics Reports 338, 1 (2000); E.G. Maksimov, M.L. Kulic, and O.V. Dolgov, Advances in Condensed Matter Physics 2010, 423725 (2010)