d-wave superconductivity induced by short-range antiferromagnetic

correlations in the Kondo lattice systems

Guang-Ming ZhangDept. of Physics, Tsinghua University, Beijing, China

Workshop on “Heavy Fermions and Quantum Phase Transitions”

10 – 12 Nov. 2012, IOP, Beijing, China

OUTLINE

• Basic physics in heavy fermion systems

• AFM order at half-filling and relation with Kondo screening effect

• FM order at small electron densities and relation with Kondo screening

• Fermi surface of heavy Fermi liquid under short-range AFM correlations

• Heavy fermion superconductivity induced by AFM short-range correlations under the Kondo screening effect

• Conclusion

Collaborator: Lu YU at Institute of Physics, Chinese Academic Sciences of China.

• Kondo physics in dilute magnetic impurities – the crossover between high T and low T At high T, free moment scatters conduction electrons → ln T resistivity. At low T, Kondo singlet/resonance forms → local Fermi liquid.

• In the Kondo lattice systems, the Kondo singlets as Landau quasiparticles leads to a large Fermi surface.

Basic physics in heavy fermion systems

Y. Onuki and T. Komastubara, J. Magnetism & Magnetic Materials, 54, 433 (1986).

• Nature of magnetically order and relation to the Kondo screening effect

One key issue:

Another key issue:

• Kondo temperature is a very high energy scale!

• Heavy Fermi liquid state is a good starting point.

• Heavy fermion SC is driven by the AFM spin fluctuations!

• Mechanism of heavy fermion superconductivity and its relation to AFM correlations

Heavy Fermi liquid state in the Kondo lattice model

Model Hamiltonian:

Fermion rep. of local moments:

Hybridization parameter

2

21

2

2, VJNf

c

J

JfcH

V

V

mfk

k

k

kkk

22

21 VJE kkk

Mean field Hamiltonian:

Renormalized band energies:

kkkk

iii sSJccH

Dramatic changes of Fermi surface due to the Kondo screening !

Small Fermi surface Large Fermi surface

Can the Kondo screening coexist with AFM long-range order?

At the half-filling, the heavy Fermi liquid becomes the Kondo insulating state.

The AFM long-range order can form at the small Kondo coupling regime.

Focus on the half-filled Kondo lattice model

AFM order parameters:(SDW like)

Kondo screening parameter: Vcddccddc iiiiiiii

Renormalized bands energies:

Longitudinal interaction -> polarization effect

Transverse interaction -> spin-flip scatterings

Both antiferromagnetic correlations and Kondo screening effect can be considered on equal footing within a mean field theory !

AFM phase

Kondo singlet phase

Coexistence phase

Ord

er

para

met

ers

The numerical calculations are performed later on a square lattice with .11 JJJ

J/t

J/t

Qua

sipa

rticl

e en

ergy

Coexistence of Kondo screening and AFM long-range order is confirmed by QMC !

……..

Abstract

Can the FM long-range order coexist with Kondo screening effect?

When the conduction electron density is far away from the half-filling, the FM long-range order can be developed in the small Kondo coupling regime.

Focus on the Kondo lattice model far away from half-filling

Order parameters:

Quasiparticle energy bands:

Mean field Hamiltonian:

Two possible FM long-range order states coexisting with Kondo screening effect

Spin non-polarized FM Spin polarized FM

The spin-polarized FM coexists with the Kondo screening has been confirmed by a recent dynamic mean field theory.

8 ,

0.17

0.05

K

C

ord B

T K

T K

M

Recent experimental discovery

Our recent results on heavy fermion ferromagnet

G. M. Zhang, et. al., in preparation.

II I

The energy gap of spin-up quasiparticles

n=0.2

n=0.2

What happens to the heavy Fermi liquidin the presence of short-range antiferromagnetic correlations ?

Dramatic changes of Fermi surface due to AFM correlations !

Heavy Fermi liquid AFM metallic state

Heisenberg exchange coupling

Kondo exchange coupling

22VJW K kkk

Kondo-Heisenberg lattice model in the limit of HK JJ

MF order parameters:

Renormalized band energies:

MF model Hamiltonian:

Two different renormalized band structures due to different types of hybridizations

Hybridization between c-electrons with f-holes

Hybridization betweenc-electrons with f-particles

0 0

,-coscos'4coscos2 yxyx kktkkt k

yxH kkJ coscosk

On a square lattice:

Ground state is unstable!

Self-consistent MF equations:

.0 withsolution the obtain always we, For HK JJ

9.0cn

Low renormalized band changes as KH JJ / Fermi surface changes as KH JJ /

Ground state energy analysis and quantum phase transitions9.0cn

Effective mass changes

9.0cn

The electron filling factor dependence of the phase transitions

HF metal phase

AFM metal phase

Can heavy fermion superconductivity be induced by short-range antiferromagnetic correlations ?

Kondo-Heisenberg lattice model in the limit of HK JJ

MF order parameters:

MF model Hamiltonian:

Kondo singlet formation

Spinon pairing attraction form

Kondo singlet pairing order parameter

Spinon-spinon pairing order parameter

The local AFM short-range correlations favor the spinon-spinon pairing with d-wave symmetryon the square lattice! yx kk coscos0 k

The ground state is a superconducting state coexisting with the Kondo screening !

Main result of the mean fieldSuperconducting pairing order parameter of the conduction electrons is induced by both the spinon-spinon pairing and a finite Kondo screening !

Heavy quasiparticle band energies: (two positive energy bands)

222241

2,

2241

2222221

1,

22,

21,1, ,

kkkk

kkk

kkkk

HK

KH

JVJE

VJJE

EEEE

Node

Gap

Spinon-spinon pairing distribution function in Brillouin zone

Conduction electron pairing distribution function in Brillouin zone

Ground state energy density and its derivative

8.0n

,0.2/

,3.0/'

c

tJ

tt

K

A quantum phase transition from nodal to nodaless superconductivity occurs!

arXiv: 1208.3684

Possible example of quasi-two dimensional heavy fermion superconductor

Conclusions• Kondo screening can coexist with the AFM order as a ground state of the

Kondo insulating phase

• Kondo screening can also coexist with the FM order in Kondo lattice model: either spin polarized or spin non-polarized phase.

• AFM short-range correlations can change the Fermi surface dramatically, leading to Lifshitz transitions

• Heavy fermion superconductivity can be driven by AFM short-range correlations under the Kondo screening effect.