Puzzling pairing in thenon-centrosymmetric superconductor LaNiC2

ISIS Facility, STFC Rutherford Appleton Laboratory

School of Physical Sciences

Puzzling pairing in the

non-centrosymmetric superconductor

LaNiC2

Jorge Quintanilla

SEPnet, University of Kent

Hubbard Theory Centre, Rutherford Appleton Laboratory

CMMP’10, University of Warwick, 15 December 2010

Adrian Hillier (RAL)

Bob Cywinski (Huddersfield)

James F. Annett (Bristol)

Bayan Mazidian (Bristol and RAL)

Collaborators:

STFC, SEPnetFunding:

LaNiC2 – a weakly-correlated, paramagnetic superconductor?

Tc=2.7 K

W. H. Lee et al., Physica C 266, 138 (1996)V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998)

ΔC/ TC=1.26 (BCS: 1.43)

specific heat susceptibility

ISIS

muSR

Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)

Relaxation due to electronic moments

Moment

size

~ 0.1G

(~ 0.01μB)

(longitudinal)

Timescale:

> 10-4s~

e

_

e

backward detector

forward detector

sample



Moment

size

~ 0.1G

(~ 0.01μB)

(longitudinal)

Timescale:

> 10-4s~

e

_

e

backward detector

forward detector

sample



Moment

size

~ 0.1G

(~ 0.01μB)

(longitudinal)

Timescale:

> 10-4s~

e

_

e

backward detector

forward detector

sample



Moment

size

~ 0.1G

(~ 0.01μB)

Spontaneous, quasi-static fields appearing at Tc

⇒ superconducting state breaks time-reversal symmetry[ c.f. Sr2RuO4 - Luke et al., Nature (1998) ]

(longitudinal)

Timescale:

> 10-4s~

e

_

e

backward detector

forward detector

sample

CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations

Symmetry of the gap function

kk

kkkˆ

See J.F. Annett Adv. Phys. 1990.

Neutron diffraction

30 40 50 60 70 800

5000

10000

15000

20000

25000

30000

35000

Inte

nsity (

arb

un

its)

2o

Orthorhombic Amm2 C2v

a=3.96 Å

b=4.58 Å

c=6.20 Å

Data from

D1B @ ILL

Note no inversion centre.

C.f. CePt3Si (1), Li2Pt3B & Li2Pd3B (2), ...

(1) Bauer et al. PRL’04 (2) Yuan et al. PRL’06


Singlet, triplet, or both?

ˆ k0 0

0 0

dx idy dz

dz dx idy

singlet

[ 0(k) even ]

triplet

[ d(k) odd ]


Singlet, triplet, or both?

Impose Pauli’s exclusion principle: , ' k ', k

Neglect (for now!) spin-orbit coupling:

ˆ k either singlet yi ˆ0', kk

or triplet yi ˆˆ.', σkdk

Singlet and triplet representations of SO(3):

Γns = - (Γn

s)T , Γnt = + (Γn

t)T


Character table


180o


C2vSymmetries and

their characters

Sample basis

functions

Irreducible

representation

E C2 v ’v Even Odd

A1 1 1 1 1 1 Z

A2 1 1 -1 -1 XY XYZ

B1 1 -1 1 -1 XZ X

B2 1 -1 -1 1 YZ Y

Character table


These must be combined with the singlet and triplet representations of SO(3).


SO(3)xC2v Gap function

(unitary)

Gap function

(non-unitary)

1A1 (k)=1 -

1A2 (k)=kxkY -

1B1 (k)=kXkZ -

1B2 (k)=kYkZ -

3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ

3A2 d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ

3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX

3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY

Possible order parameters




(unitary)

Gap function

(non-unitary)

1A1 (k)=1 -

1A2 (k)=kxkY -

1B1 (k)=kXkZ -

1B2 (k)=kYkZ -

3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ


3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX

3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY





(unitary)

Gap function

(non-unitary)

1A1 (k)=1 -

1A2 (k)=kxkY -

1B1 (k)=kXkZ -

1B2 (k)=kYkZ -

3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ


3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX

3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY





(unitary)

Gap function

(non-unitary)

1A1 (k)=1 -

1A2 (k)=kxkY -

1B1 (k)=kXkZ -

1B2 (k)=kYkZ -

3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ


3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX

3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY

Non-unitaryd x d* ≠ 0





(unitary)

Gap function

(non-unitary)

1A1 (k)=1 -

1A2 (k)=kxkY -

1B1 (k)=kXkZ -

1B2 (k)=kYkZ -

3A1 d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ


3B1 d(k)=(0,0,1)kX d(k)=(1,i,0)kX

3B2 d(k)=(0,0,1)kY d(k)=(1,i,0)kY

Non-unitaryd x d* ≠ 0

breaks only SO(3) x U(1) x T


* C.f. Li2Pd3B & Li2Pt3B,H. Q. Yuan et al. PRL’06

*



Spin-up superfluid coexisting with spin-down Fermi liquid.

Non-unitary pairing

0

00or

00

0ˆ



Non-unitary pairing

0

00or

00

0ˆ

C.f.



The A1 phase of liquid 3He.

Non-unitary pairing

0

00or

00

0ˆ

C.f.



The A1 phase of liquid 3He.

Non-unitary pairing

0

00or

00

0ˆ

C.f.

Ferromagnetic superconductors.

F. Hardy et al., Physica B 359-61, 1111-13 (2005)

[ See A. de Visser in Encyclopedia of Materials: Science and Technology (Eds.

K. H. J. Buschow et al.), Elsevier, 2010 ]


Ferromagnetic superconductors

A. de Visser in Encyclopedia of Materials: Science and Technology (Eds. K. H. J. Buschow et al.), Elsevier, 2010

But LaNiC2 is a paramagnet !


Isn’t there a more simple explanation?


yxz

zyx

iddd

didd

0

0ˆ

0

0k

The role of spin-orbit coupling (SOC)

Gap function may have both singlet and triplet components

kkorbitspin

',',

• However, if we have a centre of inversion

basis functions either even or odd under inversion

still have either singlet or triplet pairing (at Tc)

• No centre of inversion: may have singlet and triplet (even at Tc)



G = [SO(3)×Gc]×U(1)×T



G = [SO(3)×Gc]×U(1)×T



G = [SO(3)×Gc]×U(1)×T



G = [SO(3)×Gc]×U(1)×T



G = [SO(3)×Gc]×U(1)×T


G = Gc,J×U(1)×T



G = Gc,J×U(1)×T



G = Gc,J×U(1)×T




Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)

E.g. reflection through a vertical plane perpendicular to the y axis:

y

JJv CI ,2,

xy

z





y

JJv CI ,2,

xy

z





y

JJv CI ,2,

xy

z





y

JJv CI ,2,

xy

z





y

JJv CI ,2,

This affects d(k) (a vector under spin rotations).

xy

z





y

JJv CI ,2,

This affects d(k) (a vector under spin rotations).

It does not affect 0(k) (a scalar).xy

z


C2v,Jno t Gap function,

singlet component

Gap function,

triplet component

A1 (k) = A d(k) = (Bky,Ckx,Dkxkykz)

A2 (k) = AkxkY d(k) = (Bkx,Cky,Dkz)

B1 (k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)

B2 (k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)




C2v,Jno t Gap function,

singlet component

Gap function,

triplet component

A1 (k) = A d(k) = (Bky,Ckx,Dkxkykz)

A2 (k) = AkxkY d(k) = (Bkx,Cky,Dkz)

B1 (k) = AkXkZ d(k) = (Bkxkykz,Ckz,Dky)

B2 (k) = AkYkZ d(k) = (Bkz, Ckxkykz,Dkx)


None of these break time-reversal symmetry!



How could this happen?

Gap matrices evolve smoothly as SOC is turned on.

yi Â

yy ii ˆˆ.ˆˆ0 σkdkk

E.g. ( 1A1 )

( A1 )yzyxxyy ikkkkki ˆˆ.D,C,BÂ σ

for B = C = D = 0



How could this happen?Some instabilities split in two under the influence of SOC:

yz iki ˆˆ.0,,1 σE.g. ( 3A1(b) )

( B2 )0,1,0,0DC,B,A, with

ˆˆ.D,C,BÂ yxzyxzyzy ikkkkkikk σ

( B1 )0,0,1,0DC,B,A, with

ˆˆ.D,C,BÂ yyzzyxyzx ikkkkkikki

σ



Relativistic and non-relativistic instabilities: a complex relationship


spin-orbit coupling

spin-orbit coupling

1A1

1A2

1B1

1B2

3A1(a)

3A2(a)

3B1(a)

3B2(a)

A1

A2

B1

B2

A2

A1

B2

B1

3A1(b)

3A2(b)

3B1(b)

3B2(b)

B2

B1

B1

B2

A2

A1

A1

A2


Relativistic and non-relativistic instabilities: a complex relationship

singlet

Pairing

instabilities

non-unitary

triplet

pairing

instabilities

unitary

triplet

pairing

instabilities

A1 B1

3B1(b) 3B2(b)

1A11A2

3A1(a) 3A2(a)

A2 B2

1B11B2

3B1(a) 3B2(a)

3A1(b) 3A2(b)


The second (lower-Tc) instability can be symmetry-breaking because it is no longer an instability of the normal state:

3A1 (b)

(kz,ik

z,0)

B2

B1

i(0,kz,0)

(kz,0,0)

SOC

The experiments show a transition straight into the broken TRS phase

⇒ SOC must be small in LaNiC2


N.B. singlet component must be very small too.



Recap


What have we learned about LaNiC2?

Recap



Recap

Experimental observation:

the superconducting state

breaks time-reversal symmetry.



Recap




Theoretical implications:

non-unitary triplet pairing ; weak SOC ; split transition.



Recap




What do we not know yet?





Recap







Which of the four pairing symmetries?



Recap








Why non-unitary?



Recap








Why non-unitary?

Take this home:



Recap








Why non-unitary?

Take this home:

•There’s more than Rashba to noncentrosymmetric superconductors



Recap








Why non-unitary?

Take this home:

•There’s more than Rashba to noncentrosymmetric superconductors

•There’s more than strong correlations to unconventional pairing


Thanks!

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Puzzling pairing in thenon-centrosymmetric superconductor LaNiC2

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