Chiral singlet superconductivity in the weaklycorrelated metal LaPt3PPabitra Biswas 

STFC Rutherford Appleton Laboratory https://orcid.org/0000-0002-7367-5960Sudeep Ghosh  ( S.Ghosh@kent.ac.uk )

University of Kent https://orcid.org/0000-0002-3646-0629Jianzhou Zhao 

Singapore University of Technology and DesignDaniel Mayoh 

University of Warwick https://orcid.org/0000-0002-7020-3263Nikolai Zhigadlo 

University of Bern https://orcid.org/0000-0001-7322-5874Xiaofeng Xu 

Nanyang Technological UniversityChris Baines 

Paul Scherrer InstituteAdrian Hillier 

ISIS, Science and Technology Facilities Council, Rutherford Appleton Laboratory, Harwell Science andInnovation Campus, Oxfordshire, England OX11 0QX, United KingdomGeetha Balakrishnan 

Department of Physics https://orcid.org/0000-0002-5890-1149Martin Lees 

University of Warwick https://orcid.org/0000-0002-2270-2295

Article

Keywords: topological superconductors (SCs), superconductivity

Posted Date: December 29th, 2020

DOI: https://doi.org/10.21203/rs.3.rs-125751/v1

License: This work is licensed under a Creative Commons Attribution 4.0 International License.  Read Full License

Version of Record: A version of this preprint was published at Nature Communications on May 4th, 2021.See the published version at https://doi.org/10.1038/s41467-021-22807-8.

Chiral singlet superconductivity in the weakly correlated metal LaPt3P

P. K. Biswas,1, ∗ S. K. Ghosh,2, † J. Z. Zhao,3 D. A. Mayoh,4 N. D. Zhigadlo,5, 6

Xiaofeng Xu,7 C. Baines,8 A. D. Hillier,1 G. Balakrishnan,4 and M. R. Lees4

1ISIS Pulsed Neutron and Muon Source, STFC Rutherford Appleton Laboratory,

Harwell Campus, Didcot, Oxfordshire OX11 0QX, United Kingdom2School of Physical Sciences, University of Kent, Canterbury CT2 7NH, United Kingdom

3Co-Innovation Center for New Energetic Materials,

Southwest University of Science and Technology, Mianyang, 621010, China4Physics Department, University of Warwick, Coventry, CV4 7AL, United Kingdom

5Laboratory for Solid State Physics, ETH Zurich, 8093 Zurich, Switzerland6CrystMat Company, 8037 Zurich, Switzerland

7Department of Applied Physics, Zhejiang University of Technology, Hangzhou 310023,China8Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland

(Dated: December 10, 2020)

Topological superconductors (SCs) are novelphases of matter with nontrivial bulk topology.They host at their boundaries and vortex coreszero-energy Majorana bound states, potentiallyuseful in fault-tolerant quantum computation [1].Chiral SCs [2] are particular examples of topo-logical SCs with finite angular momentum Cooperpairs circulating around a unique chiral axis, thusspontaneously breaking time-reversal symmetry(TRS). They are rather scarce and usually featuretriplet pairing: best studied examples in bulk ma-terials are UPt3 and Sr2RuO4 proposed to be f-wave and p-wave SCs respectively, although manyopen questions still remain [2]. Chiral triplet SCsare, however, topologically fragile with the gap-less Majorana modes weakly protected againstsymmetry preserving perturbations in contrast tochiral singlet SCs [3, 4]. Using muon spin relax-ation (µSR) measurements, here we report thatthe weakly correlated pnictide compound LaPt3Phas the two key features of a chiral SC: sponta-neous magnetic fields inside the superconductingstate indicating broken TRS and low temperaturelinear behaviour in the superfluid density indi-cating line nodes in the order parameter. Usingsymmetry analysis, first principles band structurecalculation and mean-field theory, we unambigu-ously establish that the superconducting groundstate of LaPt3P is chiral d-wave singlet.

Cooper pairs in conventional SCs, such as the elemen-tal metals, form due to pairing of electrons by phonon-mediated attractive interaction into the most symmetrics-wave spin-singlet state [5]. In contrast, unconventionalSCs defined as having zero average onsite pairing am-plitude pose a pivotal challenge in resolving how super-conductivity emerges from a complex normal state. Theyusually require a long-range interaction [6] and have lowersymmetry Cooper pairs. A special class of unconven-tional SCs are the chiral SCs. A well established real-ization of a chiral p-wave triplet state is the A-phase of

superfluid He3 [7]. In addition to UPt3 and Sr2RuO4,the heavy fermion SC UTe2 is also proposed to be a chi-ral triplet SC [8]. The chiral singlet SCs are, however,extremely rare and are proposed to be realized withinthe hidden order phase of the strongly correlated heavyfermion SC URu2Si2 [9] and in the locally noncentrosym-metric material SrPtAs [10] with many unresolved issues.

LaPt3P is a member of the platinum pnictide family ofSCs APt3P (A = Ca, Sr and La) with a centrosymmetricprimitive tetragonal structure [11]. Its Tc = 1.1 Kis significantly lower than its other two isostructuralcounterparts SrPt3P (Tc = 8.4 K) and CaPt3P (Tc = 6.6K) [11] which are conventional Bardeen-Cooper-Schrieffer (BCS) SCs. Indications of the unconventionalnature of superconductivity in LaPt3P come both fromtheory: first principles Migdal-Eliashberg-theory [12]and experiments: very low Tc, unsaturated resistivityup to room temperature and a weak specific heat jumpat Tc [11]. The chiral nature of superconductivity ofLaPt3P with topologically protected Majorana Fermi-arc and Majorana flat-band, which we uncover here, fitsnicely with these characteristics.

Experimental resultsWe have performed a comprehensive analysis of thesuperconducting properties of LaPt3P using the µSRtechnique. Two sets of polycrystalline LaPt3P speci-mens, referred to here as sample-A (from Warwick, UK)and sample-B (from ETH, Switzerland), were synthe-sized at two different laboratories by completely differentmethods. Zero-field (ZF), longitudinal-field (LF), andtransverse-field (TF) µSR measurements were performedon these samples at two different muon facilities: sample-A in the MUSR spectrometer at the ISIS Pulsed Neutronand Muon Source, UK, and sample-B in the LTF spec-trometer at the Paul Scherrer Institut (PSI), Switzerland.

ZF-µSR measurements reveal spontaneous magneticfields arising just below Tc ≈ 1.1 K (example charcteriza-tion is shown by the zero-field-cooled magnetic suscepti-bility (χ) data for sample-B on the right axis of Fig. 1b)

2

-0.8

-0.6

-0.4

-0.2

0.0

0 3 6 9 12 150.05

0.10

0.15

0.20

0.25

χ

0.0 0.3 0.6 0.9 1.2 1.5 1.8

0.045

0.050

0.055

0.060

0.065

Temperature (K)

λ ZF (

µs-1

)

Tc = 1.1 K

ISIS

PSI

ISIS

0.075 K

1.5 K

0.15 K, LF 5 mT

1.75 K, LF 5 mT

Asy

mm

etry

Time (µs)

ZF-µSR

LF-µSR

a

b

FIG. 1. Evidence of TRS-breaking superconductivity

in LaPt3P by ZF-µSR measurements. a) ZF-µSR timespectra collected at 75 mK and 1.5 K for sample-A of LaPt3P.The solid lines are the fits to the data using Eq. 1. b) Thetemperature dependence of the extracted λZF (left axis) forsample-A (ISIS) and sample-B (PSI) showing a clear increasein the muon spin relaxation rate below Tc. The PSI datahas been shifted by 0.004 µs−1 to match the baseline valueof the ISIS data. Variation of the zero-field-cooled magneticsusceptibility (χ) on the right axis for sample-B.

associated with a TRS breaking superconducting statein both samples of LaPt3P, performed on different in-struments. Fig. 1a shows representative ZF-µSR timespectra of LaPt3P collected at 75 mK (superconduct-ing state) and at 1.5 K (normal state) on sample-A atISIS. The data below Tc show a clear increase in muon-spin relaxation rate compared to the data collected inthe normal state. To unravel the origin of the sponta-neous magnetism at low temperature, we collected ZF-µSR time spectra over a range of temperatures acrossTc and extracted temperature dependence of the muon-spin relaxation rate by fitting the data with a GaussianKubo-Toyabe relaxation function G(t) [13] multiplied byan exponential decay:

A(t) = A(0)G(t)exp(−λZFt) +Abg (1)

where, A(0) and Abg are the initial and backgroundasymmetries of the ZF-µSR time spectra, respectively.G(t) = 1

3 + 23

(

1− σ2ZFt

2)

exp(

−σ2ZFt

2/2)

. σZF and λZF

represent the muon spin relaxation rates originating from

the presence of nuclear and electronic moments in thesample, respectively. In the fitting, σZF is found tobe nearly temperature independent and hence fixed tothe average value of 0.071(4) µs−1 for sample-A and0.050(3) µs−1 for sample-B. The temperature dependenceof λZF is shown in Fig. 1b. λZF has a distinct system-atic increase below Tc for both the samples which im-plies that the effect is sample and spectrometer inde-pendent. Moreover, the effect can be suppressed veryeasily by a weak longitudinal field of 5 mT for boththe samples. It is shown in Fig. 1a for sample-A. Thisstrongly suggests that the additional relaxation below Tc

is not due to rapidly fluctuating fields [14], but ratherassociated with very weak fields which are static or qua-sistatic on the time-scale of muon life-time. The spon-taneous static magnetic field arising just below Tc isso intimately connected with superconductivity that wecan safely say its existence is direct evidence for TRS-breaking superconducting state in LaPt3P. From thechange ∆λZF = λZF(T ≈ 0) − λZF(T > Tc) we can es-timate the corresponding spontaneous internal magneticfield at the muon site Bint ≈ ∆λZF/γµ = 0.22(4) G forsample-A and 0.18(2) G for sample-B which are verysimilar to that of other TRS breaking SCs [15]. Here,γµ/(2π) = 13.55 kHz/G is the muon gyromagnetic ratio.

0 2 4 6 8 10 12 14-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

2 4 6 8 10 12 14

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.15

0.18

0.21

0.24

0.27

0.30

10.10

10.12

10.14

10.16

1.3 K

Asy

mm

etr

y

Time (µs)

a b

70 mK

Time (µs)

Temperature (K)

σ (µ

s-1)

c

Inte

rnal

field

(m

T)

Tc

FIG. 2. Superconducting properties of LaPt3P by

TF-µSR measurements. TF-µSR time spectra of LaPt3Pcollected at a) 1.3 K and b) 70 mK for sample-A in a trans-verse field of 10 mT. The solid lines are the fits to the datausing Eq. 2. c) The temperature dependence of the extractedσ (left panel) and internal field (right panel) of sample-A.

3

0.0 0.2 0.4 0.6 0.8 1.0 1.20.0

0.2

0.4

0.6

0.8

1.0 ISIS data

PSI data

s-wave

p-wave

chiral d-wave

ρ/ρ 0

Temperature (K)

FIG. 3. Evidence of chiral d-wave superconductivity

in LaPt3P. Superfluid density (ρ) of LaPt3P as a functionof temperature normalized by its zero-temperature value ρ0.The solid lines are fits to the data using different models ofgap symmetry. Inset shows the schematic representation ofthe nodes of the chiral d-wave state.

We show the TF-µSR time spectra for sample-A inFig. 2a and Fig. 2b at two different temperatures. Thespectrum in Fig. 2a shows only weak relaxation mainlydue to the transverse (2/3) component of the weak nu-clear moments present in the material in the normalstate at 1.3 K. In contrast, the spectrum in Fig. 2b inthe superconducting state at 70 mK shows higher relax-ation due to the additional inhomogeneous field distribu-tion of the vortex lattice, formed in the superconductingmixed state of LaPt3P. The spectra are analyzed usingthe Gaussian damped spin precession function [13]:

ATF (t) = A(0) exp(

−σ2t2/

2) cos (γµ 〈B〉 t+ φ)

+Abg cos (γµBbgt+ φ) . (2)

Here A(0) and Abg are the initial asymmetries of themuons hitting and missing the sample respectively. 〈B〉and Bbg are the internal and background magnetic fields,respectively. φ is the initial phase and σ is the Gaussianmuon spin relaxation rate of the muon precession signal.The background signal is due to the muons implantedon the outer silver mask where the relaxation rate ofthe muon precession signal is negligible due to very weaknuclear moments in silver. Fig. 2c shows the temper-ature dependence of σ and internal field of sample-A.σ(T ) shows a change in slope at T = Tc which keeps onincreasing with further lowering of temperature. Such anincrease in σ(T ) just below Tc indicates that the sampleis in the superconducting mixed state and the formationof vortex lattice has created an inhomogeneous field dis-tribution at the muon sites. The internal fields felt by themuons show a diamagnetic shift in the superconductingstate of LaPt3P, a clear signature of bulk superconduc-

tivity in this material.The true contribution of the vortex lattice field dis-

tribution to the relaxation rate σsc can be estimated asσsc = (σ2 − σ2

nm)1/2, where σnm = 0.1459(4) µs−1 is

the nuclear magnetic dipolar contribution assumed to betemperature independent. Within the Ginzburg-Landautheory of the vortex state, σsc is related to the Londonpenetration depth λ of a SC with high upper critical fieldby the Brandt equation [16]:

σsc (T )

γµ= 0.06091

Φ0

λ2 (T ), (3)

where Φ0 = 2.068× 10−15 Wb is the flux quantum. Thesuperfluid density ρ ∝ λ−2. Fig. 3 shows the tempera-ture dependence of ρ for LaPt3P. It clearly varies withtemperature down to the lowest temperature 70 mK andshows a linear increase below Tc/3. This nonconstantlow temperature behaviour is a signature of nodes in thesuperconducting gap.The pairing symmetry of LaPt3P can be understood

by analysing the superfluid density data using differentmodels of the gap function ∆k(T ). For a given pairingmodel, we compute the superfluid density (ρ) as

ρ = 1 + 2

⟨∫ ∞

∆k(T )

E√

E2 − |∆k(T )|2∂f

∂EdE

⟩

FS

. (4)

Here, f = 1/(

1 + eE

kBT

)

is the Fermi function and

〈〉FS represents an average over the Fermi surface (as-sumed to be spherical). We take ∆k(T ) = ∆m(T )g(k)where we assume a universal temperature dependence

∆m(T ) = ∆m(0) tanh[

1.82 {1.018 (Tc/T − 1)}0.51

]

[17]

and the function g(k) contains its angular dependence.We use three different pairing models: s-wave (singleuniform superconducting gap), p-wave (two point nodesat the two poles) and chiral d-wave (two point nodes atthe two poles and a line node at the equator as shownin the inset of Fig. 3). The fitting parameters are givenin the Supplemental Material. We note from Fig. 3that both the s-wave and the p-wave models lead tosaturation in ρ at low temperatures which is clearlynot the case for LaPt3P and the chiral d-wave modelgives an excellent fit down to the lowest temperature.Nodal SCs are rare since the SC can gain condensationenergy by eliminating nodes in the gap. Thus thesimultaneous observation of nodal and TRS-breakingsuperconductivity makes LaPt3P a unique material.

DiscussionWe investigate the normal state properties of LaPt3P bya detailed band structure calculation using density func-tional theory within the generalized gradient approxima-tion consistent with previous studies [12, 18]. LaPt3Pis centrosymmetric with a paramagnetic normal staterespecting TRS. It has significant effects of spin-orbit

4

ky

k z

-π

-π/2

0

π/2

π

-π -π/2 0 π/2 π

a

kx

ky

-π

-π/2

0

π/2

π

-π -π/2 0 π/2 π

b (001) surface BZ

(100) surface BZ

Zero energyflat band

MajoranaArc state

kxky

kz

Bulk BZ

c d

FIG. 4. Properties of the normal and superconducting states of LaPt3P. Projections of the four Fermi surfaces ofLaPt3P with SOC on the y− z plane in a and x− y plane in b. The thickness of the lines are proportional to the contributionof the Fermi surfaces to the DOS at the Fermi level (green– 10.3%, blue– 43.4%, orange– 40% and magenta– 6.3%). The pointnodes of the chiral d-wave gap are shown by red dots in a and the line node reside on the x − y plane in b. c) Schematicview of the Majorana Fermi-arc and the zero energy Majorana flat-band corresponding to the two Weyl point nodes and theline node respectively on the respective surface Brillouin zones (BZs) assuming a spherical Fermi surface. d) Berry curvatureF(k) corresponding to the two Weyl nodes on the x− z plane. Arrows show the direction of F(k) and the colour scale showsits magnitude = 2

πarctan(|F(k)|). ∆0 = 0.5µ was chosen for clarity while a more realistic weak-coupling limit ∆0 ≪ µ gives a

more sharply peaked curvature at the Fermi surface.

coupling (SOC) induced band splitting near the Fermilevel (∼ 120 meV, most apparent along the MX highsymmetry direction). Kramer’s degeneracy survives inthe presence of strong SOC due to centrosymmetry andSOC only produces small deformations in the Fermi sur-faces [19]. The shapes of the Fermi surfaces play an im-portant role in determining the thermodynamic proper-ties of the material. The projections of the four Fermisurfaces of LaPt3P on the y−z and x−y plane are shownin Fig. 4a and Fig. 4b respectively with the Fermi surfacesheets having the most projected-DOS at the Fermi levelshown in blue and orange. It shows the multi-band na-ture of LaPt3P with orbital contributions mostly comingfrom the 5d orbitals of Pt and the 3p orbitals of P.

LaPt3P has a nonsymmorphic space group P4/mmm(No. 129) with point group D4h. From the group the-oretical classification of the SC order parameters withinthe Ginzburg-Landau theory [15, 20], the only possiblesuperconducting instabilities with strong SOC which canbreak TRS spontaneously at Tc correspond to the two2D irreducible representations, Eg and Eu, of D4h [21].The superconducting ground state in the Eg channel isa pseudospin chiral d-wave singlet state with gap func-tion ∆(k) = ∆0kz(kx + iky) where ∆0 is an amplitudeindependent of k. While the Eu order parameter is apseudospin nonunitary chiral p-wave triplet state withd-vector d(k) = [c1kz, ic1kz, c2(kx + iky)] where c1 andc2 are material dependent real constants independent ofk.

We compute the quasi-particle excitation spectrumfor the two TRS breaking states on a generic singleband spherical Fermi surface using the Bogoliubov-deGennes mean field theory [15, 20]. The chiral d-wave sin-

glet state leads to an energy gap = |∆0||kz|√

k2x + k2y.

It has a line node at the “equator” for kz = 0 and

two point nodes at the “north” and “south” poles(shown in Fig. 4a). The low temperature thermo-dynamic properties are, however, dominated by theline node because of its larger low energy DOS thanthe point nodes. The triplet state has an energy

gap =√

g(kx, ky) + 2c21k2z − 2|c1||kz|

√

f(kx, ky) + c21k2z

where f(kx, ky) = c22(k2x + k2y). It has only two point

nodes at the two poles and no line nodes. Thus, the lowtemperature linear behaviour of the superfluid densityof LaPt3P shown in Fig. 3 is only possible in the chirald-wave state with a line node in contrast to the tripletstate with only point nodes which will give a quadraticbehaviour and saturation at low temperatures. ThusLaPt3P is one of the rare unconventional SCs for whichwe can unambiguously identify the superconducting or-der parameter. The point nodes and the line node for thechiral d-wave state on the Fermi surface sheets of LaPt3Pare shown in Fig. 4a and Fig. 4b.

We now discuss the topological properties of the chirald-wave state of LaPt3P based on a generic single-bandspherical Fermi surface (chemical potential µ = k2F /(2m)where kF is the Fermi wave vector and m is the elec-tron mass) [7, 22]. However, topological protection ofthe nodes also ensures stability against multiband effects.The effective angular momentum of the Cooper pairs isLz = +1 (in units of ~) with respect to the chiral c-axis.The equatorial line node acts as a vortex loop in mo-mentum space [23] and is topologically protected by a1D winding number w(kx, ky) = 1 for k2x + k2y < k2F and= 0 otherwise. The nontrivial topology of the line nodeleads to two-fold degenerate zero-energy Majorana boundstates in a flat band on the (0, 0, 1) surface BZ as shownin Fig. 4c. As a result, there is a diverging zero-energyDOS leading to a zero-bias conductance peak (which canbe really sharp [24]) measurable in STM. This inversion

5

symmetry protected line node is extra stable due to evenparity SC [3, 24]. The point nodes on the other hand areWeyl nodes and are impossible to gap out by symmetry-preserving perturbations. They act as a monopole andan anti-monopole of Berry flux as shown in Fig. 4d andare characterized by a kz dependent topological invariant,the sliced Chern number C(kz) = Lz for |kz| < kF withkz 6= 0 and = 0 otherwise (see the Supplemental Materialfor details). As a result, the (1, 0, 0) and (0, 1, 0) surfaceBZs each have a Majorana Fermi arc which can be probedby STM as shown in Fig. 4c. There are two-fold degen-erate chiral surface states with linear dispersion carryingsurface currents leading to local magnetisation that maybe detectable using SQUID magnetometry. One of thekey signatures of chiral edge states is the anomalous ther-mal Hall effect (ATHE) which depends on the length ofthe Fermi arc in this case. Impurities in the bulk can,however, increase the ATHE signal by orders of magni-tude [25] over the edge contribution making it possibleto detect with current experimental technology [26]. Wealso note that a 90◦ rotation around the c-axis for thechiral d-wave state leads to a phase shift of π/2 whichcan be measured by corner Josephson junctions [27].

Acknowledgments: PKB gratefully acknowledges theISIS Pulsed Neutron and Muon Source of the UK Science& Technology Facilities Council (STFC) and Paul Scher-rer Institut (PSI) in Switzerland for access to the muonbeam times. SKG acknowledges the Leverhulme Trustfor support through the Leverhulme early career fellow-ship. The work at the University of Warwick was fundedby EPSRC,UK, Grant EP/T005963/1. XX is partiallysupported by the National Natural Science Foundation ofChina (Grants 11974061, U1732162). NDZ thanks K. Po-varov and acknowledges support from the Laboratory forSolid State Physics, ETH Zurich where synthesis studieswere initiated.

METHODS

µSR TECHNIQUE

µSR is a very sensitive microscopic probe to detect thelocal-field distribution within a material. This techniquehas been widely used to search for very weak fields (of theorder of a fraction of a gauss) arising spontaneously in thesuperconducting state of TRS breaking SCs. The othergreat use of this technique is to measure the value andtemperature dependence of the London magnetic pene-tration depth, λ, in the vortex state of type-II SCs [28].1/λ2(T ) is in turn proportional to the superfluid densitywhich can provide direct information on the nature of thesuperconducting gap. Details of the µSR technique aregiven in the Supplemental Material.

SAMPLE PREPARATION AND

CHARACTERISATION

Two sets of polycrystalline samples (referred to assample-A and sample-B) of LaPt3P were synthesizedat two different laboratories (Warwick, UK and PSI,Switzerland) by completely different methods. While,sample-A was synthesized by solid state reaction method,sample-B was synthesized using the cubic anvil high-pressure and high-temperature technique. Details of thesample preparation and characterization are given in theSupplemental Material.

DFT CALCULATION

The first principles density functional theory (DFT)calculations were performed by the full potential lin-earized augmented plane wave method implemented inthe WIEN2k package [29]. The generalized gradient ap-proximation with the Perdew-Burke-Ernzerhof realiza-tion was used for the exchange-correlation functional.The plane wave cutoff Kmax is given by Rmt ∗ Kmax =8.0. For the self-consistent calculations, the BZ integra-tion was performed on a Γ-centered mesh of 15× 15× 15k-points.

Data availability

All the datasets that support the findings of this studyare available from the corresponding author upon reason-able request. The ISIS DOI for our MUSR source datais https://doi.org/10.5286/ISIS.E.RB1720467.

∗ pabitra.biswas@stfc.ac.uk† S.Ghosh@kent.ac.uk

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Figures

Figure 1

Evidence of TRS-breaking superconductivity in LaPt3P by ZF-uSR measurements. (see manuscript �le forfull legend)

Figure 2

Superconducting properties of LaPt3P by TF-uSR measurements. (see manuscript �le for full legend)

Figure 3

Evidence of chiral d-wave superconductivity in LaPt3P. (see manuscript �le for full legend)

Figure 4

Properties of the normal and superconducting states of LaPt3P. (see manuscript �le for full legend)

Supplementary Files

This is a list of supplementary �les associated with this preprint. Click to download.

LaPt3PSMv06.pdf