Superfluid He and Unconventional Superconductivity › ~ieremin › teaching ›...

Superfluid 3He and Unconventional Superconductivity

Introduction The Fermi-liquid normal state of 3He The pairing interaction in 3HeSuperfluid phases of 3HeUnconventional superconductors

I. Eremin, Max-Planck Institut für Physik komplexer Systeme, Dresden, Germany

Institut für Mathematische/Theoretische Physik, TU-Braunschweig, Germany

I. Eremin, MPIPKS

Introduction: Two superfluids Introduction: Two superfluids ⇒⇒ 44He and He and 33HeHe

I. Eremin, MPIPKS

Why helium is the only example of quantum fluid?

2/122⎟⎟⎠

⎞⎜⎜⎝

⎛=

TmkBdB

hπλ and typical interatomic distance d

0.270.4

0.30.07

d(nm)(nm)dBλ

Ne

He

He is essentially a quantum liquid !!

I. Eremin, MPIPKS


4He 3He

S=0 particles - bosons

Exp: Kapitza, 1938Nobel Prize 1978

Exp: Osheroff, Richardson, Lee, 1972Nobel Prize 1996

S=1/2 particles - fermions

I. Eremin, MPIPKS


Normal state

( )∑∑≠=

−+∇−=ji

jiNi

i Vm

H rr21

2ˆ

,1

22h

( )jiV rr −

van der Waals like forces

( ) 6RCaeRV bR −= −

( ) ( )NNnnNNn EH σσσσ ,,...,,,,...,,ˆ1111 rrrr Ψ=Ψ

I. Eremin, MPIPKS

Introduction: Normal state of Introduction: Normal state of 33HeHe

For 3He the many-particle wave function has to be odd under permutation of any pair spatial and spin coordinates

1) Let us put V=0 and T=0

( )

( ) ( ) ( )( ) ( ) ( )

( ) ( )NNNN

NN

NN

NNn Nσϕσϕ

σϕσϕσϕσϕσϕσϕ

σσ

rr

rrrrrr

rr

.........

...

...

!1,,...,,

11

2222112

1221111

11 =Ψ

kF ( ) 3/123 nkF π=

KF 9.4=ε FcT ε0001.0~

degenerate Fermi gas

I. Eremin, MPIPKS

FermiFermi--liquid concept of the normal state ofliquid concept of the normal state of 33HeHe

2) V(r) is strong and cannot be neglected

( )∑∑≠=

−+∇−=ji

jiNi

i Vm

H rr22

ˆ,1

22 λλ h

Turn on the interaction adiabatically

10 =⇒= λλ

)1()()0(nnn Ψ⇒Ψ⇒Ψ λ


I. Eremin, MPIPKS

a) momentum distribution

)()( λσσ

λσ ΨΨ= +

kkk ccn⎩⎨⎧

><

=F

F

kkkk

n01

σk

0=λ

1=λ

1) kF is defined at the discontinuity

2) For spherically symmetric systems position of kF remains unchanged

Luttinger theorem !

b) single-particle excited states)0(

0)0( Ψ=Ψ +

σσ kk c( )

e

F

mkk

2

222)0( −=h

kε

( )*

222)1(

2mkk F−

≈= hλεkKmm Fe 1,3* ≈= ε


I. Eremin, MPIPKS

c) two-particle excitations

)0(0''

)0('', Ψ=Ψ ++

σσσσ kkkk cc

)0('

)0(kk εε +=energy

due to interaction

( )'',' σσεε kkkk f++

( )'', σσ kkf Effective interaction between quasiparticles

( )'',21

'',''0 σσδδδε

σσσσσ

σ

kkkk

kkkk

k fnnnEE ∑∑ ++=

Total energy (Landau)

I. Eremin, MPIPKS


Distinction of quasiparticles due to the spin orientation:

( )αβαβαβ δδδ σskk .ˆ21 += nn

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ −=⎟⎟

⎠

⎞⎜⎜⎝

⎛=

1001

,0

0,

0110

zyx ii

σσσ

( ) ( )( )∑∑ ⋅+++='',

'21'0 ˆˆ',',21

σσ

δδδεkk

kkkkkk

k sskkkk ffnnnEE

( ) ( ) ( )

( ) ( ) ( )∑

∑=

=

lllF

lllF

PZfN

PFfN

ϑε

ϑε

cos',

cos',

2

1

kk

kkexpansion


I. Eremin, MPIPKS

The success of the Fermi-liquid theory in 3He is that one needs to know only few parameters: m*, F0 , F1 , Z0 , Z1

emm 3* = F0 , F1 large and positive Z0 = -3, Z1 is small

The spin susceptibility

( )4/1 0

0

Z+=

χχ

the system is close to the ferromagnetism Leggett (1975)

I. Eremin, MPIPKS

The pairing interaction in liquid The pairing interaction in liquid 33He He

Candidate 1: van der Waals potential

( ) ( ) ( ) ( ) ( )∑∫ +−− ==l

llli PVrdVeV θcos', 2

123' rkk rkk

gives too small Tc !

Candidate 2: Spin fluctuations

( ) ( ) '04

10 ˆˆ

11', kk sskk ⋅

+≈

ZZ

NV

Fε

30 −=Z 41ˆˆ ' +=⋅ kk ss

43ˆˆ ' −=⋅ kk ss

attractive for ferromagnetic fluctuations

repulsive for antiferrmomagnetic fluctuations

I. Eremin, MPIPKS

Superfluid phases Superfluid phases 33He He

BCS-like model

( ) ( )∑∑ −+−

++ +−=',

'',

',ˆkk

kkkkk

kkk kk δγβααβγδσ

σσμε ccccVccH

Mean-field approximation

( )( ) ...',ˆ',

''''int ++≈∑ −+−

+−

+−

+

kkkkkkkkkkkk δγβαδγβααβγδ ccccccccVH

Analog of BCS gap:

( ) ( ) ( )( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔΔΔ

⇒⎟⎟⎠

⎞⎜⎜⎝

⎛=

↓↓↓↑

↑↓↑↑

↓↓−↑↓−

↓↑−↑↑−

kkkk

kkkkk

kkkk

cccccccc

F

consists of triplet and singlet parts

I. Eremin, MPIPKS


to devide the triplet and singlet component αββα kkkk −− −= cccc

( ) ( )( ) ( ) ( )( ) yIi σσkd

kkkk

k ⋅+Δ=⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔΔΔ

↓↓↓↑

↑↓↑↑

( ) ( )( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( ) ( )⎟⎟⎠

⎞⎜⎜⎝

⎛++Δ−+Δ+−

=⎟⎟⎠

⎞⎜⎜⎝

⎛ΔΔΔΔ

↓↓↓↑

↑↓↑↑

kkkkkkkk

kkkk

yxz

zyx

iddddidd

scalar (singlet) component has even parity ( ) ( )kk −Δ=Δ

Vector (triplet) component has odd parity ( ) ( )kk −−= dd

Gap equation can have solutions either even or odd

I. Eremin, MPIPKS


Strong experimental evidence that the triplet pairing occur in 3He !

restrict to the small k-values

( ) ( )∑= kkk ϕθηνν ,lmlmYd

for l=1 – p-wave pairing

zyx ,,=ν

( ) ( )∑= kkk ϕθηνν ,ii pd zyx ppp ,,

In total d(k) can depend on nine parameters

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=

zzzyzx

yzyyyx

xzxyxx

i

ηηηηηηηηη

ην

ν refers to the d-orientation (spin projection)

i refers to the spatial coordinates

I. Eremin, MPIPKS


Two important phases of 3He

1) Anderson-Brinkmann-Morel (ABM) state

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

00000001 i

iνη

( ) )0,0,( yx ippd +=k2) Balian-Werthamer (BW) state

[ ]⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡=

100010001

iνη ( ) ),,( yyx pppd =k


I. Eremin, MPIPKS

( ) )0,0,( yx ippd +=k ( ) ),,( yyx pppd =k

1) Anderson-Brinkmann-Morel (ABM) state

2) Balian-Werthamer (BW) state

constant gap on the whole Fermi surfacegap vanishes at k = (0, 0, ± kF )

Superconducting gap can have zeros!

I. Eremin, MPIPKS


Excitation energy:

( ) ( ) ( ) ( ) ( )kdkdkdk k*22 ×±+−= μεE

in ABM and BW states ( ) ( ) 0* =× kdkd

( ) ( ) ( ) 22 kdk k +−= μεE Similar to the usual BCS state

Unconventional Superconductors Unconventional Superconductors

I. Eremin, MPIPKS

In 3He l=1 state means a Cooper-pair wave function which is zero if r1 = r2

kk Δ=Δ R̂Conventional superconductor

Unconventional superconductor kk Δ≠Δ R̂

For at least one symmetry operation R̂

Inversion symmetry:

( ) ( )kk −= εε

( ) ( )kk −−= dd

All triplet superconductors are unconventional

System with strong Coulomb repulsion!

Unconventional Superconductors Unconventional Superconductors

I. Eremin, MPIPKS

I. Eremin, MPIPKS

Unconventional Superconductors: Group Theory Application Unconventional Superconductors: Group Theory Application

• The microscopic mechanism in most unconventional superconductors is not yet firmly known

I. Eremin, MPIPKS


• Classification according to the group theory of the gap function

I. Eremin, MPIPKS


( ) ( )kkm

m

mfΓ

ΓΓ∑=Δ η

( ) ( )kk ννν ηm

m

mfd Γ

ΓΓ∑=

( )km

fΓ basis functions of the irreducible representation

(i) The energy eigenstate can be labelled by the irreducible representation of the corresponding point group symmetry G of the crystal

(ii) The degeneracy of the each energy state is determined by the dimension of the irreducible representation Γm

HighHigh--TTcc cuprates cuprates ⇒⇒ Group Theory ApplicationGroup Theory Application

• Crystal structure of the copper oxides: [Bednorz and

Mueller 1986]4

232 CuOSrLa ++− xx

1.5Sr2RuO4

39La1.85Sr0.15CuO4

92YBa2Cu3O7

138HgBa2Ca2Cu3O8+δ

Tc (K)Material

Singlet superconductor S=0, the point group symmetry is D4h

• Layered structure: importance of the reduced dimensionality

• Tc depends on the number of the CuO2 layers per unit cell

I. Eremin, MPIPKS

I. Eremin, MPIPKS


B2g1

B1g1

A2g1

A1g1

d mΓ ( )km

fΓ

yx kk coscos + s-wave

dx2-y2-wave

dxy-wave

( )yxyx kkkk coscossinsin +

yx kk coscos −

yx kk sinsin

I. Eremin, MPIPKS


for d-wave components superconducting gap changes sign in the first BZ

superconducting gap has points of nodes

+++

+

−−

−

−

Which of this states is realized?

( )yx kk coscos2

+Δ ( )yx kk coscos

2−

Δ ( )yx kk sinsinΔ

HighHigh--TTcc cuprates cuprates ⇒⇒ Symmetry of the order parameterSymmetry of the order parameter

I. Eremin, MPIPKS

1) Linear density of states below Δmax

HighHigh--TTcc cuprates cuprates ⇒⇒ Symmetry of the order parameterSymmetry of the order parameter

I. Eremin, MPIPKS

dx2-y2-wave state is experimentally observed

Alternative mechanisms of the CooperAlternative mechanisms of the Cooper--pairingpairing

1) Pairing from purely repulsive interaction (Kohn, Luttinger, 1965)

( )rkrrV F2cos)( 3−∝Friedel oscillations

pairing in high-angular momentum l > 0

( ){ }42exp~/ lTT Fc − too low !!

2) Pairing due to spin fluctuations (Berk and Schrieffer)

( )tSItHB

,),( rrr

h

r

μ−=

Induced spin polarization

( ) ( ) ( )','','',' 3 tHttrdtdtS B rrrr ∫ −−=rr

χμ

I. Eremin, MPIPKS

I. Eremin, MPIPKS

Alternative mechanisms of the CooperAlternative mechanisms of the Cooper--pairingpairing

Spin density-spin density interaction

( ) ( ){ } ( ) ( )',',','',''2

332

2

tStSttttrrddIHsf rrrrrrrr

h ∫ −−−−−−= χχ

Effective pairing interaction

( ) ( )kkqq −=== 'Re4

3 20 χIV S

( ) ( )kkqq −=−== 'Re4

21 χIV S

Theory I (AFI):(R.B. Laughlin, S. Sachdev, P. Lee, P.W. Anderson, T.M. Rice ... )

Theory II (FLmetallic):(J.-R. Schrieffer, D. Scalapino, A. Chubukov, D. Pines...)

I. Eremin, MPIPKS

HighHigh--TTcc cuprates cuprates ⇒⇒ possible mepossible mechanism of the Cooperchanism of the Cooper--pairingpairing

)( 42x

3x2 CuOSrLa ++

−

Possible mechanismPossible mechanism of of superconductivity for superconductivity for highhigh--TTcc cuprates cuprates ⇒⇒ spin spin fluctuation mediated fluctuation mediated CooperCooper--pairingpairing

( )sf

Qeffeff

iUUV

ωωξ

χω

−−+=)(

2AF

21,

Qqq

• d-wave symmetry and high-Tc are possible !

∑ )(Δ−

−=)(Δ'

')'(2

)'(

k

kk

kkk

EVeff

( )yx kk coscos2

0 −Δ

=)(Δ k

•Repulsive nature of the interaction

• (Monthoux, Scalapino, Millis, Monien, Pines, PRB (1989, 1991, 1994))

I. Eremin, MPIPKS

Triplet superconductivityTriplet superconductivity in Srin Sr22RuORuO44

•• SrSr22RuORuO44 isostructural to La2CuO4 ⇒ basic element RuO2-planes• superconductor at Tc= 1.5K (Y. Maeno et al., Nature (1995))

I. Eremin, MPIPKS

Pairing due to spin fluctuations?

I. Eremin, MPIPKS

Spin singlet versus spin tripletSpin singlet versus spin triplet pairingpairing

( ) ( )kkqq −=== 'Re4

3 20 χIV S

( ) ( )kkqq −=−== 'Re4

21 χIV S

Condensation energy ( )FS

cond NE 2

21

kk Δ−= ε

( )yx kk coscos2

0 −Δ ( )yxz kkd sinsin ±

( )kεN

I. Eremin, MPIPKS

Conclusions and OutlookConclusions and Outlook

(i) Unconventional superconductors give rise to a new phenomena

(ii) The theory of unconventional superconductivity is not finished

(iii) New materials are discovered in accelerating pace

HighHigh--TTcc cuprates cuprates ⇒⇒ feedback on the spin susceptibilityfeedback on the spin susceptibility

at T=0 )(1),(Im 0 QkkQkk

QkkQkkQ ++

++ −−⎟⎟⎠

⎞⎜⎜⎝

⎛ ΔΔ+−∝Ω ∑ EE

EEkωδ

εεχ

for d-wave symmetry

( ) kQk Δ−=+−+Δ=Δ + )cos()cos(0 ππ yx kk

),(Im 0 ΩQχ shows a jump at |Δk|+ |Δk+Q| = 70meV

• effect of the coherence factors

I. Eremin, MPIPKS

• via Kramers-Kronig relation real part shows a log singularity

Qq =

I. Eremin, MPIPKS

HighHigh--TTcc cuprates cuprates ⇒⇒ feedback on the spin susceptibilityfeedback on the spin susceptibility

( ) ( )202

0

0

),(Im),(Re1),(Im),(Im

Ω+Ω−Ω

=Ωqq

qqχχ

χχUU 0

1

H. Fong et al., PRB 61, 14773 (2000)

So-called resonance peak forms!

Superfluid He and Unconventional Superconductivity › ~ieremin › teaching ›...

Documents

Transcript of Superfluid He and Unconventional Superconductivity › ~ieremin › teaching ›...