SUPERCONDUCTIVITY IN SUPERCONDUCTIVITY IN SYSTEMS WITH STRONG SYSTEMS WITH STRONG

CORRELATIONSCORRELATIONS

N.M.Plakida

Joint Institute for Nuclear Research,

Dubna, Russia

Dubna, 28.08.2006

● Dense matter vs Strong correlations

● Weak and strong corelations

● Cuprate superconductors

● Effective p-d Hubbard model

● AFM exchange and spin-fluctuation pairing

● Results for Tc and SC gaps

● Tc (a) and isotope effect

Outline

Dense matter → strong correlations

Weak corelations → conventional metals, Fermi gas

Strong corelations → unconventional metals,

non-Fermi liquid

0

WH = ∑i,j σ t i,j c

+i,σ c j,σ

Hubbard model

H = ∑i,j σ t i,j c+

i,σ c j,σ

+ U ∑i ni ↑ ni ↓

where U > W

U

Weak correlations Strong correlations

2

3

1 2

3

4

Spin exchange

1

i j

Weak corelations → conventional metals, Fermi gas

One-band metal

i j

t12

2

1

0

W

Strong corelations → two Hubbard subabnds: inter-subband hopping

2

i j

1 0s

s

W

and intra - subband hopping

Structure of Hg-1201 compound ( HgBa2CuO4+δ )

After A.M. Balagurov et al.

Tc as a function of doping (oxygen O2– or fluorine F1– )

Abakumov et al. Phys.Rev.Lett. (1998)

CuO2

HgO3 doping atom

Ba

0.12 0.24

Tcmax = 96 K

px

py

dx2-y2

Electronic structure of cuprates: CuO2 plain

Undoped materials:

one hole per CuO2 unit cell.

According to the band theory:

a metal with half-filled band,

but in experiment:

antiferromagnetic insulators

Cu3d dx2-y2 orbital and O2p

(px , py) orbitals:

Cu 2+ (3d9) – O 2– (2p6)

Electronic structure of

cuprates Cu 2+ (3d 9) – O 2–

(2p 6)

Band structure calculations predict

a broad pdσ conduction band:

half-filled antibonding 3d(x2 –y2) - 2p(x,y) band

Mattheis (1987)

px

py

dx2-y2

d

d+ p

d+Ud

1

2

EFFECTIVE HUBBARD p-d MODEL

Model for CuO2 layer:

Cu-3d (εd) and O-2p (εp) states,

with Δ = εp − εd ≈ 3 eV,

Ud ≈ 8 eV, tpd ≈ 1.5 eV,

Cell-cluster perturbation theory

Exact diagonalization of the unit cell Hamiltonian Hi(0)

gives new eigenstates:

E1 = εd - μ → one hole d - like state: l σ > E2 = 2 E1 + Δ → two hole (p - d) singlet state: l ↑↓ >

Xiαβ = l iα > < iβ l with

l α > = l 0 >, l σ >= l ↑ >, l ↓ >, and l 2 >= l ↑↓ >Hubbard operators rigorously obey the constraint: Xi

00 + Xi↑↑ + Xi

↓↓ + Xi 22 = 1

― only one quantum state can be occupied at any site i.

We introduce the Hubbard operators for these states:

In terms of the projected Fermi operators: Xi

0σ = ci σ (1 – n i – σ) , Xi σ 2 = ci – σ n i σ

Commutation relations for the Hubbard operators: [Xi

αβ , Xj γμ ]+(−) = δi j δ β γ

Xiαμ

Here anticommutator is for the Fermi-like operators as Xi0σ, Xi

σ 2,

and commutator is for he Bose-like operators as Xiσ σ, Xi

22, e.g. the spin operators: Si

z = (1/2) (Xi++ – Xi

– , – ), S i

+ = Xi+ –, Si

– = Xi– + ,

or the number operator

Ni = (Xi++ + Xi

– –) + 2 Xi22

These commutation relations result in the kinematic interaction.

The two-subband effective Hubbard model reads:

Kinematic interaction for the Hubbard operators:

Dyson equation for GF in the Hubbard model We introduce the Green Functions:

Equations of motion for the matrix GF are solved within the Mori-type projection technique:

The Dyson equation reads:

with the exact self-energy as the multi-particle GF:

Mean-Field approximation

– anomalous correlation function – SC gap for singlets.

→ PAIRING at ONE lattice site but in TWO subbands

Frequency matrix:

where

Equation for the pair correlation Green function gives:

Gap function for the singlet subband in MFA :

is equvalent to the MFA in the t-J model

For the singlet subband: μ ≈ ∆ and E2 ≈ E1 ≈ – ∆ :

i j

t12

2

1

0

W

AFM exchange pairing

All electrons (holes) are paired in the conduction band.

Estimate in WCA gives for Tcex :

Self-energy in the Hubbard model

SCBA:

, where

Self-energy matrix:

Xj Xmtij tml

Bi Bl

tij tmlGjm

il≈

Gap equation for the singlet (p-d) subband:

where the kernel of the integral equation in SCBA

defines pairing mediated by spin and charge fluctuations.

2

i j

10

s

s

W

Spin-fluctuation pairing

Estimate in WCA gives for Tcsf :

Equation for the gap and Tc in WCA

The AFM static spin susceptibility

where ξ ― short-range AFM correlation length, ωs ≈ J ― cut-off spin-fluctuation energy.

Normalization condition:

Estimate for Tc in the weak coupling approximation

Wehave:

Fig.1. Tc ( in teff units):(i)~spin-fluctuation pairing, (ii)~AFM exchange pairing , (iii)~both contributions

NUMERICAL RESULTS

Parameters: Δpd / tpd = 2, ωs / tpd = 0.1, ξ = 3, J = 0.4 teff, teff ≈ 0.14 tpd ≈ 0.2 eV,

tpd = 1.5 eV

↑

δ=0.13

N.M. Plakida, L. Anton, S. Adam, and Gh. Adam,

JETP 97, 331 (2003). Exchange and Spin-Fluctuation Mechanisms of Superconductivity in Cuprates.

Fig. 2. ∆(kx, ky)

( 0 < kx, ky < π) at optimal doping δ ≈ 0.13

Unconventional d-wave pairing:

∆(kx, ky) ~∆ (coskx - cosky)

Large Fermi surface (FS)

FS

Tc (a) and pressure dependenceFor mercury compounds, Hg-12(n-1)n, experiments show dTc / da ≈ – 1.35·10 3 (K /Å), or d ln Tc / d ln a ≈ – 50 [ Lokshin et al. PRB 63 (2000) 64511 ]

From

Tc ≈ EF exp (– 1/ Vex ), Vex = J N(0) , we get: d ln Tc / d ln a = (d ln Tc / d ln J) · (d ln J / d ln a) ≈ – 40 , where J ≈ tpd

4 ~ 1/a14

Isotope shift: 16 O → 18 O

Isotope shift of TN = 310K for La2CuO4 , Δ TN ≈ −1.8 K

[ G.Zhao et al., PRB 50 (1994) 4112 ]

and αN = – d lnTN /d lnM ≈ – (d lnJ / d lnM) ≈ 0.05

Isotope shift of Tc : αc = – d lnTc / d lnM =

= – (d lnTc / dln J) (d lnJ/d lnM ) ≈ (1/ Vex) αN ≈ 0.16

For conventional, electron-phonon superconductors,

d Tc / d P < 0 , e.g., for MgB2, d Tc / d P ≈ – 1.1 K/GPa, while for cuprates superconductors, d Tc / d P > 0

CONCLUSIONS

Superconducting d-wave pairing with high-Tc

mediated by the AFM superexchange and spin-fluctuations is proved for the p-d Hubbard model.

Retardation effects for AFM exchange are suppressed:

∆pd >> W , that results in pairing of all electrons (holes) with high Tc ~ EF ≈ W/2 .

Tc(a) and oxygen isotope shift are explained. The results corresponds to numerical solution to the

t-J model in (q, ω) space in the strong coupling limit.

In the limit of strong correlations, U >> t, the Hubbard model

can be reduced to the t-J model by projecting out the upper band:

Comparison with t-J model

Jij = 4 t2 / U is the exchange energy for the nearest neighbors,

in cuprates J ~ 0.13 eV

In the t-J model the Hubbard operators act only in the subspace of

one-electron states: Xi0σ = ci σ (1 – n i – σ)

We consider matrix (2x2) Green function (GF) in terms of Nambu operators:

The Dyson equation was solved in SCBA for the normal G11 and

anomalous G12 GF in the linear approximation for Tc calculation:

The gap equation reads

where interaction

Fig.1. Spectral function for the t-J model in the symemtry direction

Γ(0,0) → Μ(π,π) at doping δ = 0.1 (a) and δ = 0.4 (b) .

Numerical results

1. Spectral functions A(k, ω)

Fig.2. Self-energy for the t-J model in the symemtry direction

Γ(0,0) → Μ(π,π) at doping δ = 0.1 (a) and δ = 0.4 (b) .

2. Self-energy, Im Σ(k, ω)

Fig.3. Electron occupation numbers for the t-J model in the quarter of BZ, (0 < kx, ky < π) at doping δ = 0.1 (a) and δ = 0.4 (b) .

3. Electron occupation numbers N(k) = n(k)/2

Fig.4. Fermi surface (a) and the gap Φ(kx, ky) (b) for the t-J model in the quarter of BZ (0 < kx, ky < π) at doping δ = 0.1.

4. Fermi surface and the gap function Φ(kx, ky)

CONCLUSIONS

Superconducting d-wave pairing with high-Tc

mediated by the AFM superexchange and spin-fluctuations is proved for the p-d Hubbard model.

Retardation effects for AFM exchange are suppressed:

∆pd >> W , that results in pairing of all electrons (holes) with high Tc ~ EF ≈ W/2 .

Tc(a) and oxygen isotope shift are explained. The results corresponds to numerical solution to the

t-J model in (q, ω) space in the strong coupling limit.

Electronic structure of cuprates: strong electron correlations Ud

> Wpd

At half-filling 3d(x2 –y2) - 2p(x,y) band splits into UHB and

LHB Insulator with the charge-transfer gap: Ud > Δ = εp

− εd

Publications:

N.M. Plakida, L. Anton, S. Adam, and Gh. Adam,

Exchange and Spin-Fluctuation Mechanisms of Superconductivity in

Cuprates. JETP 97, 331 (2003). N.M. Plakida , Antiferromagnetic exchange mechanism

of superconductivity in cuprates. JETP Letters 74, 36 (2001) N.M. Plakida, V.S. Oudovenko,

Electron spectrum and superconductivity in the t-J model at moderate doping.

Phys. Rev. B 59, 11949 (1999)

Outline

Effective p-d Hubbard model AFM exchange pairing in MFA Spin-fluctuation pairing Results for Tc and SC gaps Tc (a) and isotope effect Comparison with t-J model

WHY ARE COPPER–OXIDES THE ONLY HIGH–Tc SUPERCONDUCTORS with Tc > 100 K?

Cu 2+ in 3d9 state has the lowest 3d level in transition metals

with strong Coulomb correlations Ud >Δpd = εp – εd.

They are CHARGE-TRANSFER INSULATORS

with HUGE super-exchange interaction J ~ 1500 K —>

AFM long–range order with high TN = 300 – 500 K

Strong coupling of doped holes (electrons) with spins

Pseudogap due to AFM short – range order

High-Tc superconductivity

EFFECTIVE HUBBARD p-d MODEL

Model for CuO2 layer:

Cu-3d ( εd ) and

O-2p (εp ) states

Δ = εp − εd ≈ 2 tpd

In terms of O-2p Wannier states

d

d+ p

d+Ud

1

2