Puzzling pairing in the non-centrosymmetric superconductor LaNiC2

ISIS Facility, STFC School of
Rutherford Appleton Laboratory Physical Sciences

Puzzling pairing in the
non-centrosymmetric superconductor
LaNiC2
Jorge Quintanilla
SEPnet, University of Kent
Hubbard Theory Centre, Rutherford Appleton Laboratory

Collaborators: Adrian Hillier (RAL)
Bob Cywinski (Huddersfield)
James F. Annett (Bristol)
Bayan Mazidian (Bristol and RAL)

Funding: STFC, SEPnet

CMMP’10, University of Warwick, 15 December 2010

LaNiC2 – a weakly-correlated, paramagnetic
superconductor?
W. H. Lee et al., Physica C 266, 138 (1996)
V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998)

specific heat susceptibility

ΔC/TC=1.26
Tc=2.7 K (BCS: 1.43)

Relaxation due to electronic moments _


e 

e
(longitudinal) sample
forward
detector
backward
detector

Moment
size Timescale:
~ 0.1G > 10-4s
~
(~ 0.01μB)

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

Relaxation due to electronic moments _


e 

e
(longitudinal) sample
forward
detector
backward
detector

Moment
size Timescale:
~ 0.1G > 10-4s
~
(~ 0.01μB)

Spontaneous, quasi-static fields appearing at Tc
⇒ superconducting state breaks time-reversal symmetry
[ c.f. Sr2RuO4 - Luke et al., Nature (1998) ]
PRL 102 117007 (2009)

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

Symmetry of the gap function

  k    k 
k   
ˆ

   k    k 

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

Neutron diffraction
35000

30000
Data from
25000 D1B @ ILL
Intensity (arb units)

20000

15000

10000

5000
Orthorhombic Amm2 C2v
0
30 40 50 60 70 80 a=3.96 Å
b=4.58 Å
o
2 
c=6.20 Å

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?

 0  0   x  id y
d dz 
k  
ˆ   

 0 0   dz dx  id y 
singlet triplet
[ 0(k) even ] [ d(k) odd ]


Singlet, triplet, or both?
Neglect (for now!) spin-orbit coupling:

Singlet and triplet representations of SO(3):
Γns = - (Γns)T , Γnt = + (Γnt)T

Impose Pauli’s exclusion principle:  , ' k   ', k
 k
ˆ either singlet  , ' k    0 k i y
ˆ
or triplet  , ' k   dk .σi y
ˆ ˆ



Character table

180o

PRL 102 117007 (2009)


Character table
C2v Symmetries and Sample basis
their characters functions
Irreducible E C2 v ’v Even Odd
representation
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

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

PRL 102 117007 (2009)


Possible order parameters
SO(3)xC2v Gap function Gap function
(unitary) (non-unitary)
1A
1 (k)=1 -
1A
2 (k)=kxkY -
1B
1 (k)=kXkZ -
1B
2 (k)=kYkZ -
3A d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
1
3A d(k)=(0,0,1)kXkYkZ d(k)=(1,i,0)kXkYkZ
2
3B d(k)=(0,0,1)kX d(k)=(1,i,0)kX
1
3B d(k)=(0,0,1)kY d(k)=(1,i,0)kY
2

PRL 102 117007 (2009)


1A
1 (k)=1 -
1A
2 (k)=kxkY -
1B
1 (k)=kXkZ -
1B
2 (k)=kYkZ -
3A d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
1
2
3B d(k)=(0,0,1)kX d(k)=(1,i,0)kX
1
3B d(k)=(0,0,1)kY d(k)=(1,i,0)kY
2

Non-unitary
d x d* ≠ 0
PRL 102 117007 (2009)


1A
1 (k)=1 -
1A
2 (k)=kxkY -
breaks only SO(3) x U(1) x T *
1B
1 (k)=kXkZ -
1B
2 (k)=kYkZ -
3A d(k)=(0,0,1)kZ d(k)=(1,i,0)kZ
1
2
3B d(k)=(0,0,1)kX d(k)=(1,i,0)kX
1
3B d(k)=(0,0,1)kY d(k)=(1,i,0)kY
2

* C.f. Li2Pd3B & Li2Pt3B,
H. Q. Yuan et al. PRL’06
Non-unitary
d x d* ≠ 0
PRL 102 117007 (2009)


Non-unitary pairing
Spin-up superfluid
coexisting with spin- ˆ     0  0 0 
 or 
down Fermi liquid.    0  0   
 0   


Non-unitary pairing
Spin-up superfluid
 or 
 0   

C.f.


Non-unitary pairing
Spin-up superfluid
 or 
 0   

C.f.

The A1 phase of
liquid 3He.


Non-unitary pairing
Spin-up superfluid
 or 
 0   

C.f.

The A1 phase of
liquid 3He.
Ferromagnetic
F. Hardy et al., Physica B 359-
61, 1111-13 (2005) superconductors.
[ See A. de Visser in Encyclopedia of
Materials: Science and Technology (Eds.
K. H. J. Buschow et al.), Elsevier, 2010 ]


Ferromagnetic superconductors

But LaNiC2 is a paramagnet !

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


Isn’t there a more simple explanation?


The role of spin-orbit coupling (SOC)
 , ' k   spin ' orbit k 
 ,

Gap function may have both singlet and triplet components

 0  0   d x  id y dz 
k   
ˆ
 d
  0 0  z d x  id y 

• 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 = Gc,J×U(1)×T


z
E.g. reflection through a vertical
plane perpendicular to the y axis:
 v, J  I C2y, J

y x

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


z
 v, J  I C2y, J

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

y x



z
 v, J  I C2y, J

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

It does not affect 0(k) (a scalar).
y x




C2v,Jno t Gap function, Gap function,
singlet component 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, Gap function,
singlet component 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?
k    0 k i y  dk .σi y
ˆ ˆ ˆ ˆ

Gap matrices evolve smoothly as SOC is turned on.

E.g. Ai y
ˆ ( 1A1 )

ˆ  ˆ ˆ
 A i y  Bk y , Ck x , Dk x k y k z .σ i y ( A1 )
for B = C = D = 0



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

E.g. 1, i,0k z .σi y
ˆ ˆ ( 3A1(b) )


ˆ  
Ak y k z i y  Bk z , Ck x k y k z , Dk x .σ i y 
ˆ ˆ
 ( B2 )
 with A, B, C, D  0,1,0,0 

 i
ˆ  
Ak x k z i y  Bk x k y k z , Ck z , Dk y .σ i y 
ˆ ˆ
 ( B1 )
 with A, B, C, D  0,0,1,0 



Relativistic and non-relativistic
instabilities: a complex relationship
1A A1
1
1A A2
2
3A (b)
B2
1B 1
1 B1 B1
1B B2 B1
2 3A (b)
2
B2
3A (a)
A2 3B (b)
A2
1
1
3A (a) A1 A1
2
A1
3B (a)
1
B2 3B (b)
2
A2
3B (a)
2
B1
spin-orbit coupling

spin-orbit coupling



Relativistic and non-relativistic
instabilities: a complex relationship
A1 A2 B1 B2

singlet
Pairing 1A 1A 1B 1B
instabilities 1 2 1 2

non-unitary
3B 3A
triplet 1(b) 1(b)
pairing 3B 3A
instabilities 2(b) 2(b)

unitary
triplet
3A 3A 3B 3B
pairing
1(a) 2(a) 1(a) 2(a)
instabilities


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

B2 The experiments show a
(kz,0,0) transition straight into the
3A (b)
1 broken TRS phase
(k ,ik ,0) B1 ⇒ SOC must be small in
z z
i(0,kz,0)
LaNiC2
N.B. singlet component must
be very small too.
SOC



Recap


Recap
What have we learned about LaNiC2?


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!

Puzzling pairing in the non-centrosymmetric superconductor LaNiC2

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Puzzling pairing in the non-centrosymmetric superconductor LaNiC2