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Contributed talk at CMMP 2010 (university of Warwick, December 2010)
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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
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
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
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
Relaxation due to electronic moments
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Singlet, triplet, or both?
ˆ k0 0
0 0
dx idy dz
dz dx idy
singlet
[ 0(k) even ]
triplet
[ d(k) odd ]
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Character table
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
180o
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
These must be combined with the singlet and triplet representations of SO(3).
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
Non-unitaryd x d* ≠ 0
Possible order parameters
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
Non-unitaryd x d* ≠ 0
breaks only SO(3) x U(1) x T
Possible order parameters
* C.f. Li2Pd3B & Li2Pt3B,H. Q. Yuan et al. PRL’06
*
Hillier, Quintanilla & Cywinski,PRL 102 117007 (2009)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Spin-up superfluid coexisting with spin-down Fermi liquid.
Non-unitary pairing
0
00or
00
0ˆ
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Spin-up superfluid coexisting with spin-down Fermi liquid.
Non-unitary pairing
0
00or
00
0ˆ
C.f.
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Spin-up superfluid coexisting with spin-down Fermi liquid.
The A1 phase of liquid 3He.
Non-unitary pairing
0
00or
00
0ˆ
C.f.
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Spin-up superfluid coexisting with spin-down Fermi liquid.
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 ]
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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 !
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Isn’t there a more simple explanation?
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
G = [SO(3)×Gc]×U(1)×T
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
G = Gc,J×U(1)×T
The role of spin-orbit coupling (SOC)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
G = Gc,J×U(1)×T
The role of spin-orbit coupling (SOC)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
G = Gc,J×U(1)×T
The role of spin-orbit coupling (SOC)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
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,
This affects d(k) (a vector under spin rotations).
xy
z
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
The role of spin-orbit coupling (SOC)
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,
This affects d(k) (a vector under spin rotations).
It does not affect 0(k) (a scalar).xy
z
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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)
The role of spin-orbit coupling (SOC)
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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)
The role of spin-orbit coupling (SOC)
None of these break time-reversal symmetry!
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
How could this happen?
Gap matrices evolve smoothly as SOC is turned on.
yi ˆA
yy ii ˆˆ.ˆˆ0 σkdkk
E.g. ( 1A1 )
( A1 )yzyxxyy ikkkkki ˆˆ.D,C,BˆA σ
for B = C = D = 0
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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,BˆA yxzyxzyzy ikkkkkikk σ
( B1 )0,0,1,0DC,B,A, with
ˆˆ.D,C,BˆA yyzzyxyzx ikkkkkikki
σ
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Relativistic and non-relativistic instabilities: a complex relationship
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
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
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
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
The role of spin-orbit coupling (SOC)
N.B. singlet component must be very small too.
Quintanilla, Hillier, Annett and Cywinski, PRB 82, 174511 (2010)
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Recap
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
What do we not know yet?
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
What do we not know yet?
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
Which of the four pairing symmetries?
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
What do we not know yet?
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
Which of the four pairing symmetries?
Why non-unitary?
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
What do we not know yet?
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
Which of the four pairing symmetries?
Why non-unitary?
Take this home:
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
What do we not know yet?
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
Which of the four pairing symmetries?
Why non-unitary?
Take this home:
•There’s more than Rashba to noncentrosymmetric superconductors
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
What have we learned about LaNiC2?
Recap
Experimental observation:
the superconducting state
breaks time-reversal symmetry.
What do we not know yet?
Theoretical implications:
non-unitary triplet pairing ; weak SOC ; split transition.
Which of the four pairing symmetries?
Why non-unitary?
Take this home:
•There’s more than Rashba to noncentrosymmetric superconductors
•There’s more than strong correlations to unconventional pairing
CMMP’10, Warwick, 15 Dec 2010 blogs.kent.ac.uk/strongcorrelations
Thanks!