Post on 04-Oct-2020
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)