Session: Superconductivity Title: "LaNiC2 and LaNiGa2: The Compelling Case for Non-unitary Triplet Pairing" Authors: James F Annett (1), Gábor Csire (1,2), Sudeep Kumar Ghosh (3), Martin Gradhand (1), Adrian D Hillier (4), Jorge Quintanilla (3), Tian Shang (5,6), Michael Smidman (7) Balázs Újfalussy (8) , Philip Whittlesea (3), and Huiqiu Yuan (7,9) (1) University of Bristol (2) ICN2 (3) University of Kent (4) ISIS Facility (5) PSI (6) Universität Zürich (7) Zhejiang University (8) Hungarian Academy of Sciences (9) Nanjing University Abstract: There is a compelling case that non-centrosymmetric LaNiC2 and centrosymmetric LaNiGa2 belong to a new class of fully-gapped, internally-antisymmetric, non-unitary triplet (INT) superconductors. I will review the theoretical and experimental evidence [1], starting from the original observations of increased muon spin relaxation rate and symmetry-based arguments in favour of non-unitary triplet pairing and culminating in the current, quantitative, single-parameter theory [2]. I will highlight what remains unknown about these systems and point out possible future directions. References: [1] S. Ghosh, M. Smidman, T. Shang, J. Annett, A. Hillier, J. Quintanilla, and H. Yuan, “Recent progress on superconductors with time-reversal symmetry breaking”, Journal of Physics: Condensed Matter 33, 033001 (2021). DOI: 10.1088/1361-648X/abaa06. [2] S. Ghosh, G. Csire, P. Whittlesea, J. Annett, M. Gradhand, and J. Quintanilla, “Quantitative theory of triplet pairing in the unconventional superconductor LaNiGa2”, Phys. Rev. B 101, 100506(R), DOI: 10.1103/PhysRevB.101.100506.

1. The UK’s European university
Jorge Quintanilla / LaNiC2 and LaNiGa2:
The Compelling Case for Non-unitary
Triplet Pairing
CMQM 2021
FUNDING:
* UK EPSRC (EP/P007392/1, EP/P00749X/1)
* Hungarian NRDIO (contract K115632)
James F Annett (1)
Gábor Csire (1,2)
Sudeep Kumar Ghosh (3)
Martin Gradhand (1)
Adrian D Hillier (4)
Jorge Quintanilla (3)
Tian Shang (5,6)
Michael Smidman (7)
Balázs Újfalussy (8)
Philip Whittlesea (3)
Huiqiu Yuan (7,9)
(1) University of Bristol
(2) ICN2 (Barcelona)
(3) University of Kent
(4) ISIS Facility
(5) PSI
(6) Universität Zürich
(7) Zhejiang University
(8) Hungarian Academy of Sciences
(9) Nanjing University
[1] S. Ghosh, M. Smidman, T. Shang, J. Annett, A. Hillier,
J. Quintanilla, and H. Yuan, “Recent progress on
superconductors with time-reversal symmetry breaking”,
Journal of Physics: Condensed Matter 33, 033001 (2021).
[2] S. Ghosh, G. Csire, P. Whittlesea, J. Annett, M. Gradhand,
and J. Quintanilla, “Quantitative theory of triplet pairing in the
unconventional superconductor LaNiGa2”,
Phys. Rev. B 101, 100506(R).

2. research.kent.ac.uk/PQM
CMQM 2021
https://tinyurl.com/7wa7cvx3

3. research.kent.ac.uk/PQM
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Time-reversal symmetry breaking: LaNiC2 and LaNiGa2
LaNiGa2
AD Hillier, J Quintanilla,
and R Cywinski,
PRL 102, 117007
(2009)
LaNiC2
AD Hillier, J Quintanilla,
B Mazidian, JF Annett,
and R Cywinski,
PRL 109, 097001
(2012)
T (K)
T (K)

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Time-reversal symmetry breaking: LaNiC2 and LaNiGa2
LaNiGa2
AD Hillier, J Quintanilla,
and R Cywinski,
PRL 102, 117007
(2009)
LaNiC2
AD Hillier, J Quintanilla,
B Mazidian, JF Annett,
and R Cywinski,
PRL 109, 097001
(2012)
T (K)
T (K)
single
crystals
LaNiC2
S Sundar, SR Dunsiger, et al.
PRB 103, 014511 (2021)

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J Chen, L Jiao, J L Zhang, Y Chen, L Yang,
M Nicklas, F Steglich, and H Q Yuan,
NJP 15, 53005 (2013)
LaNiC2
(polycrystals)
Two full gaps
ZF Weng, JL Zhang, M Smidman, T Shang,
J Quintanilla, JF Annett, M Nicklas, GM Pang,
L Jiao, WB Jiang, Y Chen, F Steglich,
and HQ Yuan, PRL 117, 027001 (2016)
LaNiGa2
Penetration depth Penetration depth
LaNiC2
(single crystals)
S Sundar et al.
PRB 103, 014511 (2021)
TF-mSR

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Low-symmetry crystal ==> only 4 possible superconducting ground states
breaking time-reversal symmetry
(c.f. Sr2RuO4: 22 possibilities)
==> broken time-reversal symmetry requires
Non-unitary triplet pairing: symmetry arguments
^
Δ=(Δ↑↑ 0
0 Δ↓↓
)Γ ( k ) with Δ↑↑≠ Δ↓↓
LaNiC2: J Quintanilla, AD Hillier, JF Annett, and R Cywinski, PRB 82, 174511 (2010)
LaNiGa2: AD Hillier, J Quintanilla, B Mazidian, JF Annett, and R Cywinski, PRL 109, 097001 (2012)
(nonunitary
triplet
pairing)

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Non-unitary triplet pairing: smoking gun
Symmetry-dictated coupling to magnetisation:
Leads to sub-dominant (linear in Tc-T) magnetisation:
AD Hillier, J Quintanilla, B Mazidian, JF Annett,
and R Cywinski, PRL 109, 097001 (2012);
K Miyake, JPSJ 83, 053701 (2014)
A Sumiyama et al.,
JPSJ 84, 13702 (2015)
Prediction ... … confirmation (bulk SQUID)
AD Hillier, J Quintanilla, B Mazidian, JF Annett
& R Cywinski, PRL 109, 097001 (2012)

8. research.kent.ac.uk/PQM
CMQM 2021
Low-symmetry crystal ==> only 4 possible superconducting ground states
breaking time-reversal symmetry
(c.f. Sr2RuO4: 22 possibilities)
==> broken time-reversal symmetry requires
Non-unitary triplet pairing: symmetry arguments
^
Δ=
(Δ↑↑ 0
0 Δ↓↓
)Γ(k) with Δ↑↑≠Δ↓↓
ZF Weng, JL Zhang, M Smidman, T Shang, J Quintanilla, JF Annett, M Nicklas, GM Pang, L Jiao, WB Jiang, Y Chen,
F Steglich & HQ Yuan, PRL 117, 027001 (2016)
(nonunitary
triplet
pairing)

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The INT*
state (* Internally-antisymmetric, Nonunitary Triplet pairing state)
Must involve two different orbitals A,B
ZF Weng et al., PRL 117, 027001 (2016)
SK Ghosh,G Csire, P Whittlesea, JF Annett, M Gradhand, B Újfalussy & J Quintanilla, PRB 101, 100506(R) (2020)
Assume negative-U, equal-spin
interaction (e.g. from Hund on Ni):
2D
↑↑
2D
↓↓

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The INT*
state (* Internally-antisymmetric, Nonunitary Triplet pairing state)
ZF Weng et al., PRL 117, 027001 (2016)
SK Ghosh,G Csire, P Whittlesea, JF Annett, M Gradhand, B Újfalussy & J Quintanilla, PRB 101, 100506(R) (2020)
Postulate nearly-degenerate bands: Consistent with DFT band structures:
DJ Singh, PRB 86, 174507 (2012)
s

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Simple model: phase diagram
SK Ghosh et al., PRB 101, 100506(R) (2020)
Phase diagram can be obtained by unconstrained minimisation of F.

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Detailed model
Bogoliubov-de Gennes Theory
Bogoliubov-de Gennes Theory
(mean field)
Describe different pairing scenarios
at phenomenological level
Density Functional Theory
Density Functional Theory
(LDA)
(LDA)
Realistic calculation of the band
structure, and Fermi surface
LDA + BdG
LDA + BdG
Material specific, realistic calculations
with different pairing models
+
LaNiC2: G Csire, B Újfalussy & JF Annett, EPJB 91, 217 (2018)
LaNiGa2: SK Ghosh, G Csire, P Whittlewsea, JF Annett, M Gradhand, B Ujfalussy & JQ, PRB 101, 100506(R) (2020)

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Detailed model
Kohn-Sham Bogoliubov-de Gennes Equations: Oliviera, Gross, Kohn, PRL 60, 2430
(1988)
Self-consistency: Suvasini, Temmerman, Győrffy, PRB 48, 1202 (1999)
LDA (first principles)
Mean-field (phenomenological)
=L
Resolve L in orbital basis (L,s)
and treat as adjustable parameter
Possible within a KKR approach
G Csire et al., PRB 97, 024514 (2018)
Fully-relativistic, electrostatic Veff(r) + exchange Beff(r)

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Detailed model: pairing scenarios and coupling strength
DOS for different
pairing scenarios in
LaNiGa2.
We assumed
interactions between
two Ni d-orbitals.
Only z2
↔ xy gives
fully-gapped
spectrum.
Choosing
L(dz
2
-dxy) = 0.65eV
Reproduces
Tc = 2.1 K.
SK Ghosh,G Csire, P Whittlesea, JF Annett, M Gradhand, B Újfalussy & J Quintanilla, PRB 101, 100506(R) (2020)

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Detailed model: specific heat and density of states
Excellent fit to experimental
specific heat
Experiment
Weng et al
PRL (2016)
Theory
(this work)
Prediction of tunnelling
conductance showing clearly two
peaks in the DOS
NOTE: NO FITTING PARAMETERS!
SK Ghosh,G Csire, P Whittlesea, JF Annett, M Gradhand, B Újfalussy & J Quintanilla, PRB 101, 100506(R) (2020)
LaNiGa2

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Detailed model: magnetic moment
67%
33%
SK Ghosh,G Csire, P Whittlesea, JF Annett, M Gradhand, B Újfalussy & J Quintanilla, PRB 101, 100506(R) (2020)
LaNiGa2

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LaNiC2
Detailed model: specific heat and density of states
Similar agreement was found for LaNiC2:
– Experiment
Chen et al. NJP (2016)
Theory: – Ni dxz-dzy – Ni dz2-dxy – Ni dz2-dx2-y2
G Csire, B Ujfalussy & JF Annett, EPJB 91, 217 (2018)

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LaNiC2 and LaNiGa2 in summary
●
Broken time-reversal symmetry is very well
established experimentally (ZF-mSR)
●
Non-unitary triplet pairing is strongly supported
by theory (symmetry) and experiment (SQUID)
●
Fully-gapped “two-band” spectrum is very well
established experimentally (l,Cv,TF-mSR)
●
INT state explains all of the above naturally
●
INT-based theory describes these materials
quantitatively with a single adjustable parameter
●
Observation of the spin-resolved DOS would
confirm these as the best-understood triplet
superconductors

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www.kent.ac.uk
Thanks!