2010 Quintanilla Loughborough

ISIS Facility, STFC School of
Rutherford Appleton Laboratory Physical Sciences

Non-unitary triplet pairing in the
non-centrosymmetric superconductor
LaNiC2
Jorge Quintanilla
University of Kent
& Rutherford Appleton Laboratory

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

Funding: STFC, SEPnet

Research seminar, University of Loughborough, 26 May 2010

Photo: Eddie Hui-Bon-Hoa, www.shiromi.com




Unconventional superconductors

Photo: Kenneth G. Libbrecht, snowflakes.com







Photo: commons.wikimedia.org




‘Unconventional’
superconductors:
Photo: commons.wikimedia.org

HTSC, Sr2RuO4,
PrOs4Sb12, UPt3,
(UTh)Be13 , ...

LaNiC2
V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998)
W. H. Lee et al., Physica C 266, 138 (1996)

specific heat

ΔC/TC=1.26
(BCS: 1.43)

Tc=2.7 K

Zero field muon spin relaxation

_


e 
sample
e
forward
detector
backward
detector

Zero field μSR at ISIS
0.30

0.25
LaNiC2
0.20

3.0 K
Asymmetry

0.15

50 mK
0.10

0.05

0.00
0 5 10 15 20

Time (s)

 2
1 
  2 t 2 
Gz (t)  A0     t exp 
1 2 2
  exp t   Abckgrd

3 3
  2  



0.30

0.25
LaNiC2
0.20

3.0 K
Asymmetry

0.15

50 mK
0.10

0.05

0.00
0 5 10 15 20

Time (s)

 2
1 
  2 t 2 
Gz (t)  A0     t exp 
1 2 2

3 3
  2  
Nuclear contribution
(Kubo-Toyabe)


0.30

0.25
LaNiC2
0.20

3.0 K
Asymmetry

0.15

50 mK
0.10

0.05

0.00
0 5 10 15 20

Time (s)

 2
1 
  2 t 2 
Gz (t)  A0     t exp 
1 2 2

3 3
  2  
Nuclear contribution Electronic
(Kubo-Toyabe) contribution


Relaxation due to nuclear moments

0.10

Nuclear relaxation rate σ is
0.08
constant as a function of
temperature.
0.06
(⇒ static muons)
s-1)

0.04
A0, Abckg also T-independent.
0.02 The only T-dependence is
in .
0.00
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Temperature (K)

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)

Symmetry of the gap function

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

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

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

Assume SOC very small:

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

Assume SOC very small:  , ' k   
spin
 , ' 
orbit
k 

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

Assume SOC very small:  , ' k   
spin
 , ' 
orbit
k 
Impose Pauli’s exclusion principle:

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

Assume SOC very small:  , ' k     k 
spin
 , '
orbit

Impose Pauli’s exclusion principle:  , ' k   ', k



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

spin
 , '
orbit

  k either singlet
ˆ



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

spin
 , '
orbit

  k either singlet  , ' k    0 k i y
ˆ ˆ



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

spin
 , '
orbit

ˆ ˆ
or triplet


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

spin
 , '
orbit

ˆ ˆ
or triplet  , ' k   dk .σi y
ˆ ˆ


Character table

PRL 102 117007 (2009)

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

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 -
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

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

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

But LaNiC2 is a paramagnet !
V. K. Pecharsky, L. L. Miller, and Zy, Physical Review B 58, 497 (1998)
W. H. Lee et al., Physica C 266, 138 (1996)

Pauli paramagnet
(constant
susceptibility)

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 

• Centre of inversion:
basis functions either even or odd under inversion
 still have either singlet or triplet pairing
• No centre of inversion: may have singlet and triplet

• Simplest noncentrosymmetric system: a surface. Gor'kov & Rashba,
PRL, 87, 037004 (2001)
• Rashba term in the Hamiltonian:

• Split Fermi surface:
 k    k for spin 
 k   
 k    k for spin 
• In general, form & strength of SOC depend on details of electronic structure.
• There’s a zoo of phenomenologies for noncentrosymmetric superconductors:
Triplet: CePt3Si
Singlet (conventional): Li2Pd3B, BaPtSi3, Re3W
Singlet-triplet admixture: Li2Pt3B

• Electronic structure of LaNiC2:

Laverock, Haynes, Utfeld & Dugdale, Subedi & Singh, Hase & Yanagisawa,
PRB, 80, 125111 (2009) PRB 80, 092506 (2009) JPSJ 78, 084724 (2009)

• Electronic structure of LaNiC2:

Laverock, Haynes, Utfeld & Dugdale, Subedi & Singh, Hase &
PRB, 80, 125111 (2009) PRB 80, 092506 (2009)
Yanagisawa,
• Hase & Yanagisawa in particular have estimated JPSJ 78,
 ~ 3.1 mRy ~ ħwD ~ 400 K >> Tc = 2.7 K 084724 (2009)
• This is ~ ½ the value for CePt3Si for which Tc = 0.4 - 0.7 K. Hasegawa & Taniguchi,
JPSJ 78, 074717 (2009)
• But in CePt3Si there seems to be a pure spin triplet state, with SOC acting as pair-breaker.
• In fact de Haas-van Alphen oscillations fail to detect SOC splitting. Hashimoto et al., JPCM
16, L287 (2004)

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, arXiv :1005.1084

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 in the absence of SOC can be described in terms of
limiting cases of those in the presence of SOC.

E.g. Ai y
ˆ ( 1A1 )

ˆ  
 A i y  Bk y , Ck x , Dk x k y k z .σ i y
ˆ ˆ ( 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
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

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


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


Summary
What have we learned about LaNiC2?

Summary
 Experimental observation:
the superconducting state
breaks time-reversal symmetry.

Summary

Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.

Summary


What do we not know yet?

Summary


 Which pairing symmetry?

Summary



Why non-unitary?

Summary



Why non-unitary?

Take this home:

Summary



Why non-unitary?

Take this home:
There’s more than Rashba to
noncentrosymmetric superconductors

2010 Quintanilla Loughborough

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