Coexistence of Triple Nodal Points, Nodal Lines, and Unusual Flat Bandin intermetallic APd3 (A=Pb, Sn)

Kyo-Hoon Ahn1, Warren E. Pickett2,∗ and Kwan-Woo Lee1,3†1Department of Applied Physics, Graduate School, Korea University, Sejong 30019, Korea

2Department of Physics, University of California, Davis, California 95616, USA3Division of Display and Semiconductor Physics, Korea University, Sejong 30019, Korea

(Dated: March 18, 2018)

We investigate the electronic structure and several properties, and topological character, of thecubic time-reversal invariant intermetallic compounds PbPd3 and SnPd3 using density functionaltheory based methods. This class of compounds has a dispersionless band along the Γ − X line,forming the top of the Pd 4d bands and lying within a few meV of the Fermi level EF . Effectsof the flat band on transport and optical properties have been inspected by varying the dopingconcentration treated with the virtual crystal approximation for substitution on the Pb site. In theabsence of spin-orbit coupling (SOC), we find that surface states emerge and lead to Dirac conesalong the Γ − X directions, with Fermi arcs connecting the M -centered electronic Fermi surfacewith small hole pockets distributed around M points in the two-dimensional Brillouin zone (BZ).SOC removes degeneracy in most of the zone, providing a topological index Z2=1 on the kz = 0plane and indicating a strong topological character on that plane. The isovalent and isostructuralcompound SnPd3 shows only minor differences in its electronic structures, so it is expected displaysimilar electronic, transport, and topological properties.

PACS numbers:

I. INTRODUCTION

For the last decade one of the most active issues incondensed matter physics is topological character of theelectronic structure and the new properties that mayarise. Various types of topological matters have beenproposed and some have received experimental support.In addition to topological insulators, three-dimensional(3D) topological semimetals (TSM) have attracted largeinterest due to their unusual boundary states,[1] whichmay stimulate novel directions in electronics and spin-tronics separate from topological insulators. Recently,a novel triple point fermonic phase, having no highenergy counterparts, has been also proposed in bothnonsymmorphic[2] and symmorphic structures.[3] Thisphase, predicted along the symmetry line with 3-fold ro-tation and mirror (C3v) symmetries in a few systems ofWC-type or half-Heusler structures,[4–7] is expected tohave various unconventional properties and is experimen-tally observed in MoP.[8]

The intermetallic palladium-lead phase diagram in-cludes various compounds.[9, 10] Among them, the so-called ideal zvyagintsevite compound PbPd3 (cubicCu3Au structure) has been investigated as a catalystfor electrochemical oxygen reduction.[11–13] It has beenknown that its sister compounds can absorb a largeamount of hydrogen,[14, 15] making them of interest ascandidates for hydrogen storage or membrane separation.

Separately, flat bands in part or all of the BZ have

∗Electronic address: pickett@physics.ucdavis.edu†Electronic address: mckwan@korea.ac.kr

piqued interest for a variety of reasons. A weakly dis-persive band leads to a peak in density of states (DOS),typically much stronger and perhaps narrower than struc-tures arising from van Hove singularities. When a DOSpeak is close to, or at, the Fermi level EF , instabili-ties of the simple Fermi liquid states are encouraged;peaks can enhance a superconducting critical tempera-ture or induce Stoner-type magnetic instability.[16–18] InPbPd3 a remarkably flat band appears along the Γ −Xlines, arising from a lack of ddδ hopping such as occursin (cubic) perovskite-like systems.[18, 19] Because of thelimited phase space where the band is flat, it leads totwo-dimensional-like step in the DOS (see below). It be-comes of importance because it lies within a few meVof the Fermi energy EF , implying that transport, ther-modynamic, and infrared properties will display unusualdependences on temperature, doping, or other changes inthe system (viz. strain).

II. APPROACH

We have performed density functional theory calcula-tions based on the exchange-correlation functional of thePerdew-Burke-Ernzerhof generalized gradient approxi-mation (GGA)[20] with the all-electron full-potentialcode wien2k.[21] Since Pb is a heavy atom, SOC is in-cluded in all calculations unless otherwise noted. Figure1 (a) shows the cubic Cu3Au crystal structure (spacegroup: Pm3m, No. 221), with experimental lattice pa-rameters of a = 4.035 A for PbPd3[12, 22] and a = 3.971A for SnPd3[23, 24]. Both experimental parameters aresmaller by about 2% than our values optimized in GGA.Here, our calculations were based on the experimental

Γ X

X

X

M

M

R

M

X

Γ

M

X(a) (b)

FIG. 1: (a) Cubic L21 crystal structure of APd3, having aPd6 octahedron. (b) The bulk and (001) surface Brillouinzones (BZs) with high symmetry points. The unfilled circlesdenote positions of spinless triple nodal points, appearing inthe range of –0.3 eV to 0.2 eV.

values, unless mentioned otherwise.The topological properties were explored by the Wan-

nier function approach. From the band structures ob-tained from wien2k, Wannier functions and the corre-sponding hopping amplitudes were generated using thewannier90 [25] and wien2wannier [26] programs. Thesurface spectral functions were calculated by the wan-niertools code.[27]

The transport and optical properties were calculatedby two extensions of wien2k. Based on the semiclas-sical Boltzmann transport theory with a constant scat-tering time approximation, the transport calculationswere carried out with the boltztrap code.[28] Calcu-lation of optical properties including SOC is available inthe optics.[29] Assuming an inverse scattering lifetimeγ = 10 meV in the intra-band contribution, the dielectricfunction ε(ω), contributed by both intra- and inter-bandexcitations, was calculated. The electronic energy lossfunction is given by –Im[ε−1(ω)].

In wien2k, the Brillouin zone (BZ) was sampled bya very dense k-mesh 40×40×40 due to its sharp DOSstructure near EF . For a careful check of the position ofthe flat band, a high value of basis set cutoff RmtKmax =9 was used to determine the basis size with atomic radiiof 2.5 bohr for the both ions.

III. ELECTRONIC AND TOPOLOGICALCHARACTERS OF PbPd3

A. Electronic structure of PbPd3

The band structures of both GGA and GGA+SOCnear EF are shown in Figs. 2(a) and 2(b). In this energyregime, the bands have mostly Pb 6s, 6p above EF andPd 5d character below. The corresponding density ofstates (DOS) with Pd 5d orbital-projected DOS is givenin Fig. 2 (c).

The band structure shows unusual aspects relating totransport, thermodynamic, and topological properties.Most noticeably, a very flat band of Pd 5d character ap-pears along the Γ − X line only a few meV above theFermi level EF . The lack of dispersion reflects the lackof nearest neighbor (NN) Pd ddδ hopping. This char-acteristic appears commonly in cubic perovskite-relatedsystems,[19] but is unusual in intermetallic compounds.As indicated in Fig. 2(a), for the Γ − X line along the(100) direction, the flat band has solely the dyz orbitalcharacter of Pd2 ( 1

2120) and Pd3 ( 1

20 12 ) ions. Signifi-

cantly, in PbPd3 this flat band lies only 6 meV aboveEF , thereby giving an effective Fermi energy for holesof E∗F,h= 6 meV and corresponding Fermi temperature

T∗F,h=70 K. As can be seen in Fig. 2(c), structure in theDOS is a 2D-like step discontinuity, and electron dopingof only ∼0.01 carrier/f.u. would be required to move EFup to the band edge. Above the edge, the DOS is verysmall (semimetal-like) up to 0.2 eV.

Another aspect that we will follow is an occurrenceof 3-fold degeneracies along symmetry lines, which arebecoming known as triple nodal points (TNPs). In theabsence of SOC, as shown in Fig. 2(a), spinless TNPsΓ±4 appear at the Γ point. Along the Γ−R line, havingthe C3v symmetry, the Γ−4 band at 0.1 eV splits into adoublet Λ3 of a positive effective mass and a singlet Λ1

of a negative effective mass. The Γ+4 band just above EF

splits into a doublet Λ3 and a singlet Λ2 with negativeeffective masses in both. As previously suggested,[2, 3]the Λ3 and Λ1 crossing forms a TNP at (0.39,0.39,0.39) 2π

aat ∼0.15 eV along the Γ − R. At the R-point, a TNPof Γ+

4 is at –0.32 eV. This band splits into a doublet Λ5

of a positive effective mass and a singlet Λ2 along theR −M line, having C4v symmetry. This doublet formsa TNP with the Λ4 at ( 1

2 ,12 , 0.312) 2π

a at ∼0.19 eV. TheTNP along the C4v symmetry line has not been discussedpreviously. One may expect surface states connectingthese spinless TNPs in the absence of SOC. This happensin PbPd3, as discussed below. In additional to TNPs,there are nodal points at ∼ 0.5 eV at ( 1

2 ,12 , 0.3983) 2π

aalong the R −M line, leading to topological nodal linksshown in Fig. 3(c). These R-centered links are similarto what was proposed in the antiperovskite Cu3PdN.[33]

To probe the origin of these TNPs, the cubic structurehas been distorted, conserving the volume. Tetragonaldistortion leads to breaking C3v symmetry, thereby re-moving TNPs along the (111) direction. Both the C3v

and C4v symmetries are broken by orthormbic distor-tion, resulting in the vanishing all of the TNPs alongthe R−M line as well as along the 100 direction (seethe Supplemental Material).[38] So, it is clear that theseTNPs are prected by the mirror plus three-fold (or four-fold) rotational symmetries.

Comparison of the two band structures without andwith SOC in Fig. 2 reveals several effects of SOC. SOCleads to anticrossing of (nearly) degenerate bands as wellas lowering the flat band toward EF . At the Γ-point, two6-fold degenerate bands Γ±4 (including spin) split into

2

X Γ R M -1

-0.5

0

0.5

1E

- E

F (

eV

)

1Γ

1

−

1Γ

4

−

3Γ

4

−

3Γ

4

+

1Γ

2

+

2Γ

3

+

3Γ

4

+

1Γ

1

+

1Γ

2

−

2Γ

5

−

1Γ

4

+

1Λ

1

2Λ

3

1Λ

21Λ

11Λ

1 1Λ

2

1Λ

1

2Λ

5

1Λ

4

1Λ

2

2Λ

3

2Λ

5

1Λ

21Λ

12Λ

5

2Λ

3

X Γ

0

10

20

(a)

X Γ R M -1

-0.5

0

0.5

1

E -

EF (

eV

)

2Γ

6

−

2Γ

7

−

4Γ

8

−

4Γ

8

+

2Γ

6

+

2Γ

7

+

2Γ

6

−

4Γ

8

+

2Γ

7

−

2Γ

6

+

4Γ

8

+

2Γ

6

+

2Γ

7

+

2Γ

7

−

2Γ

6

−

2Λ

6

2Λ

7

2Λ

6

2Λ

7

2(Λ

5+Λ

6)

2Λ

4

2Λ

4

2Λ

4

2Λ

4

2(Λ

5+Λ

6)

2Λ

4

2Λ

6

2Λ

6

2Λ

6

2Λ

7

2Λ

7

2(Λ

5+Λ

6)

X Γ

0

10

20

(b)

0 4

(c)

FIG. 2: Blowup (a) GGA and (b) GGA+SOC band structures with point group representations along the X −Γ−R−M linearound the Fermi energy EF , which is set to zero. The superscripts of the notations denote degeneracies of each band. At thehigh symmetry points, the ± symbols indicate parities of each band. The Γ−R line has a C3v symmetry, while the other linesdo a C4v symmetry. The insets show the flat band lying a few meV above EF along the Γ−X line. The symmetry points ofthe band structures are given in Fig. 1(b). The X-point is the zone boundary along the (100) direction. (c) The GGA+SOCdensity of states (DOS) of PbPd3, in units of states/eV. The (yellow) shaded region corresponds to the orbital-resolved DOSof Pd 5d states. In (a), the dyz characters of Pd2 ( 1

2120) and Pd3 ( 1

20 12) ions are highlighted by thick (green) lines. Note that

there is no contribution of Pd1 (0 12

12) ion to the flat band along this direction.

FIG. 3: (Color online) GGA+SOC Fermi surfaces of (a)PbPd3 and (b) SnPd3. For PbPd3, the R-centered spherecontains electrons, while the others enclose holes. In (a), tenDirac points in the BZ appear between the sphere and thecarrot-like hole pockets. R-centered sphere. In SnPd3, largecarrots appear along the (111) directions. (c) Nodal line iso-countours connecting the Dirac points, denoted by black dots,at ∼0.5 eV along the M − R −M ′ line in GGA. The centerof the fractional BZ is the R-point.

doublet and quartet, resulting in a SOC gap of 0.3 eVdue to a band inversion. Remarkable impacts of SOCresulting in changes of surface states appear in the spin-less TNPs along Γ−R and Γ−M lines near the R-pointat ∼0.2 eV. As discussed by Zhu and coworkers,[3] theseTNPs are not proteced in the cubic structure, when in-cluding SOC. Instead, SOC driven anticrossings at theseTNPs lead to ten fourfold degenerate Dirac points at∼0.2 eV along the Γ − R and at ∼0.15 eV the R −M .A similar behavior along a C6v line was suggested in thelayered hexagonal AlB2-type metal diborides.[39] Thusthe band structure is nearly gapped at 0.2 eV, strength-ening the semimetal viewpoint of the near-EF electronicstructure of PbPd3. In fact, the electronic structure isgapped everywhere at E≈0.1 eV except for a single banddispersing upward from the zone corner R point; SOChas resulted in an anticrossing of two bands in the sameregion.

The result of this band structure is delicate Fermi sur-faces (FS). Figure 3 displays the physical Fermi surfaces

(SOC included), consisting of three types. Narrow cylin-drical open hole FSs lie along the (100) axes, intersect-ing at the zone center; these tubes are related to theflat band along Γ − X discussed above. An R-centeredelectron spheroid FS has a radius of 0.26(πa ), containing∼0.03 electrons per formula unit. Additionally, there aretiny elliptical hole pockets around this spheroid along the100 and 111 directions.

Most results obtained for our optimized lattice param-eter are very close to those for the experiment parameterpresented above, as expected from the fact that differ-ence is small (less than 2%). There is however a dis-tinction worth mentioning. For the optimized (smaller)volume, the flat band is shifted to –5 meV relative toEF . Thus modest pressure can be used to tune this bandedge through the chemical potential for the stoichiomet-ric composition. The position of the flat band can be alsotuned by both tetragonal or orthrombic strain (see theSupplemental Material).[38]

B. Topological character of PbPd3

Using a Wannier representation of the bands, we havecarried out surface calculations based on the Green func-tion approach.[30] For PbPd3, there are two possible(001) surface terminations, one containing both Pb andPd atoms (Pb-Pd) and the other containing only Pd sur-face atoms (Pd2). Figure 4 shows the surface spectralfunctions for the each termination, in both GGA andGGA+SOC. Around 0.2 eV above EF the surface zoneis gapped except around the M point, arising from thebulk bands around R that project onto M . In the ab-sence of SOC, in Figs. 4(a) and (c), the Pd2 terminationshows a weakly dispersive surface band in both directionsfrom Γ. For the Pb-Pd terminated surface only surfaceresonances within the bulk bands appear along symme-

3

FIG. 4: The (001) Surface spectral functions for (top) Pd2 and (bottom) Pb-Pd terminations within (left) GGA and (middle)GGA+SOC. The reddish regions denote projections of bulk states. The right panels display the spin texture at the two-dimensional Dirac cone appearing in (d) at Γ, having opposite helical texture in the (e) upper and (f) lower cones. The surfaceprojected symmetry points are given in Fig. 1(b).

try lines. Several surface states connecting TNPs emergeat the in-gap and inside the bulk states in both termina-tions. The positions of TNPs on the surface BZ are givenin Fig. 1(b). At Γ the points are near EF , and are at∼ −0.3 at the M . Note also that a surface state crossingEF , nearly identical in the both terminations, emergesinside the bulk states around the M point. This Fermilevel crossing band is unaffected by SOC.

Inclusion of SOC leads to a large gap just above EF atthe Γ point, resulting in changes in the surface spectrum.As shown in Fig. 4(b), in the Pd2 termination the surfacestate splits with a 0.3 eV gap around Γ point, but orbitaldegeneracy survives at the generic momenta around 0.2eV due to the presence of inversion symmetry. As shownin Figs. 4(b) and (d), SOC splits this surface band intoa pair extending from Γ to the surface zone boundaries.For Pb-Pd termination, where no surface bands aroundΓ appear within the gap in GGA, SOC produces a newoccurrence: a surface band Dirac point emerges at Γ at0.25 eV above EF (see Fig. 4(d)). This surface crossingis enabled by SOC opening a bulk gap at Γ. In the Pb-Pdtermination, a 2D Dirac cone at the Γ point is a dominantfeature. As shown in 4 (e) and (f), its spin texture showsopposite helical direction in the upper and lower cones,indicating topological character.

C. Fermi level spectral density

The surface Fermi level spectral density contours arepictured in Fig. 5. As expected from differences be-tween the two terminations in the surface spectrum, aclear distinction appears along the Γ − X line, whereasfeatures around M are nearly identical. In the Pd2 termi-nation, it leads to four clear and large Dirac cones alongthe Γ − X lines in both without and with SOC, shownin Fig. 5 (a) and (b). The bulk R point and its Fermispheroid project onto the M point, and eight protrusionsconnected to this spheroid appear for both terminationsand only show some quantitative change with SOC, ap-parently related to the bulk Fermi pockets around theR point. (For the blowup figures, see the SupplementalMaterial).[38] This coexistence of Dirac cones and Fermiarcs is similar to what was proposed in time-reversal in-variant Weyl semimetals.[34]

Calculations of the hybrid Wannier charge cen-ters(WCCs) allow a “theoretical spectroscopy” of topo-logical character.[31, 32] In this technique, the Bloch-to-Wannier transformation is applied only to one direction.These calculations were performed to establish topologi-cal character in the wanniertools code.[27] In the ap-proach, the Z2 number is calculated from the number ofcrossings of WCCs mod 2 by an arbitrary line in half

4

FIG. 5: Spectral densities of the surface Fermi contours of thesurface states shown in Fig. 4, for (top) Pd2 and (bottom)Pb-Pd terminations. The left panels are for GGA only, theright panels show results for GGA+SOC. The strong yellowishlines denote surface bands.

0.0

0.5

1.0

0.0 0.1 0.2 0.3 0.4 0.5

WC

C

k

FIG. 6: Plot of the hybrid Wannier charge centers (HWCCs)plot (red, thick lines) across a half of the Brillouin zone in thekz = 0 plane, showing an odd number of crossings between thecharge center and largest gap among two adjacent HWCCs.The gap function is designated by the blue (thin) line. Themagnitude of the wave vector ky in the horizontal axis is givenin the unit of π/a.

of BZ due to the gauge of time-reversal symmetry.[31]Practically, instead of an arbitrary line, a line betweentwo adjacent WCCs is chosen for numerical efficiency.

In PbPd3, inclusion of SOC separates bands over mostof the BZ, band crossings surviving around the R point.Thus the electronic structure on the basal planes (viz.kz = 0) can be considered as insulating, thus provid-ing a well-defined Z2 number, which can be calculatedusing the hybrid Wannier charge center method.[31] Asshown in Fig. 6, there is an odd number of crossingsof the largest gap with the charge centers, giving Z2=1.

X Γ R M

0

50

100

E -

EF (

meV

)

x = 0.00

x = 0.01

x = 0.02

x = 0.03x = 0.033

FIG. 7: Enlargement of the band structure in GGA+SOC,with horizontal lines denoting the Fermi level as amount ofelectron doping is done. The results are from the virtual crys-tal approximation applied to (Pb1−xBix)Pd3. 1% doping putsthe Fermi level precisely on the flat band.

This number is also obtained from parities of all occu-pied eigenstates at time-reversal invariant momenta inthe kz = 0 plane. The parities are given by –1 at Γ and+1 at X and M , for a product of –1, again indicatingZ2=1. Thus, this system is a strong topological insula-tor on the kz = 0 plane.

D. Strain effects

We have considered tetragonal and orthrombic distor-tions to investigate effects of strain. The Z2 indices arewell defined in both kz=0 and π/c planes in both caseseven for a small strain. Unfortunately, the correspond-ing indices are 0;(0,0,0), indicating a topologically trivalphase (see the Supplemental Material).[38] This result isdue to change of parity at Γ.

Two sister compounds of semimetallic SnPt3 andPbPt3 are worthy of note. In contrast to the Pd analogieswith several TNPS above EF , the bands are fully filled,so SOC leads to well separated bands around EF in allof the BZ.[40] Recently, the SnPt3 was proposed as aweak topological insulator,[41], while the PbPt3, with itsadditional complexities, will be discussed elsewhere.[40]

IV. ROLE OF THE FLAT BAND; ELECTRONDOPING

We have varied the electron concentration in(Pb1−xBix)Pd3, which is experimentally accessible,[22]to inspect the role of the flat band in transport and op-tical properties, using the virtual crystal approximation,for which this is an optimal application. We will focuson GGA+SOC results, which are realistic for Bi substi-tution for Pb. There is negligible change in band struc-ture, and Fig. 7 displays the rising of the Fermi level withadded electrons. The flat band precisely lies at EF for

5

200 400 600T [K]

-0.5

0

0.5

1

1.5R

H(T

) [1

0-2

cm3/C

] x = 0.00x = 0.01x = 0.02x = 0.03x = 0.033

200 400 600T [K]

0.2

0.3

0.4

σ/τ

[1

020(Ω

m s

)-1]

200 400 600T [K]

-40

0

40

80

S(T

) [µ

V/c

m]

-0.4 -0.2 0 0.2 0.4µ [eV]

-40

0

40

80

S(µ

;T)

[µV

/cm

]400 K

300 K

200 K

150 K

(a)

(c) (d)

(b)

FIG. 8: As varying doped-electron concentration,temperature-dependent (a) Hall coefficients RH(T), (b)electronic conductivity σ over scattering time τ , and (c)Seebeck coefficients S for GGA+SOC in the range of 150 K– 800 K. (d) Chemical potential µ-dependent Seebeck coef-ficients S(µ;T ) for various temperatures in GGA+SOC. In(d), the vertical dashed lines indicate the chemical potentialat each concentration, in the order of x = 0 to 0.033 fromthe left.

x = 0.01, and by x = 0.02 the intersecting cylinder FSsdisappear. At x = 0.033, the hole pockets at the Γ arefilled. Since transport and optical properties are derivedfrom the bands εk, its derivatives, and the Fermi levelposition,[42] one may anticipate specific features relatedwith the flat band in PbPd3.

A. Transport properties

The transport properties have been calculated basedon quasiclassical Bloch-Boltzmann transport theory,[42]assuming a constant scattering time τ approximation.Our results including SOC are pictured in Fig. 8. TheHall coefficient RH(T ), given in Fig. 8(a), shows substan-tial variation with temperature even at these low dopinglevels. At zero doping x=0, it is net (from competingFermi surfaces) electron-like in sign. With increasing T ,RH(T ) monotonically decreases, and just above 800 Kcrosses zero, where electron and hole contributions com-pensate. With increasing doping, RH(T ) crosses zeroat lower temperatures, e.g., for x = 0.03 exact com-pensation of holes and electrons occurs at 400K. Forx = 0.033, RH(T ) is negative for the whole range of T,indicating dominance of hole carriers. Recall that in a

multiband system RH(T ) has no simple relation to car-rier densities.[43]

Figure 8(b) shows the T -dependent conductivity overscattering time σ/τ . For a typical metal with largeN(EF ), σ/τ is proportional toN(µ)〈v2F 〉[44] with insignif-icant T -dependence. Here N(µ) and 〈v2F 〉 are the DOS atthe chemical potential µ(T ) and the thermal average ofsquare of the Fermi velocity at µ(T ), respectively. Withstructure in N(E) around EF , µ becomes T-dependent.As T rises above 400 K (x=0), σ/τ increases, by about25% at 800K, due to the sharp drop in N(E) 6 meVabove EF . As doping increases, the low temperaturevalue drops and the temperature increase is enhancedas EF approached the DOS singularity.

Related variations are reflected in the Seebeck coeffi-cient S(T, x),[44] given by

S =π2

3

k2BT

e

(dlnσ(ε)

dε

)µ

. (1)

As T increases, S(T, x) increases up to about 300-500Kdepending on x and then drops, as displayed in Fig. 8(c).For x=0.033, S(T, x) is very low, a condition that is fa-vorable for certain applications. Fig. 8(d) presents adifferent viewpoint, plotting S(µ, T ) versus µ at four val-ues of T. At each temperature, the maximum occurs atµ=0, followed by a steep drop trough zero around µ=0.1eV, before leveling off beyond µ=0.2 eV. Measurement ofthe Seebeck coefficient, which is independent of scatter-ing in the constant scattering time approximation, couldbe useful in determining the stoichiometry of PbPd3 atthe 1% level.

Although interesting features are visible in the x, µ,and T dependences of the transport coefficients, there isnothing that is as striking as might have been expectedgiven the anomalous N(E) very near EF . A contributingfeature is that the FSs are small and varied in size andshape, and the Fermi velocities are small: on the crossedcylinders being ∼ 6.4× 106 cm/sec.

B. Optical properties.

The electron energy loss function L(ω), given in Fig.9, was calculated from the imaginary part of the inversedielectric function ε(ω). The dielectric function includesboth intra- and interband contributions. In the intrabandpart, containing the Drude term, an inverse scatteringtime γ is chosen to be 10 meV.

The overall behavior of L(ω) is non-monotonically de-pendent on electron-doping level. PbPd3is a nearly trans-parent (semi)metal below ω = 75 meV. At zero doping,a peak at 0.2 eV when SOC is neglected is split by SOC,leaving peaks at 0.11, 014, and 0.27 eV. The plasmonpeak around 0.1 eV is mainly due to excitation at theΓ-point and has the strongest intensity at x = 0.03. Theintensity and position of these peaks depends somewhaton the choice of γ.

6

0 1 2ω (eV)

0

2

4

6

8L

(ω)

(x 1

0-2

)

0 0.1 0.2 0.3 0.40

2

4

x = 0.00

x = 0.01

x = 0.02

x = 0.03

x = 0.033

x = 0.00 w/o SOC

FIG. 9: Energy loss function L(ω) versus energy ω (~=1),for the various electron-doping levels. A scattering rate γ =~/τ of 10 meV has been assumed, which primarily affects thewidths of peaks but also determines the (small) intrabandcontribution. Inset: Enlarged plot below ω = 0.4, showingthe low energy plasmonic peak(s) clearly. For x=0, the resultwithout SOC is included to indicate the influence of SOC inthe loss function.

Γ X M Γ R-1

0

1

E -

EF (

eV)

Γ X M Γ R

Γ X-20

0

20

(meV

)

0 4

Γ X-20

0

20

(meV

)

(a) GGA (b) GGA+SOC (c)

FIG. 10: (Color online) Correspondence of SnPd3 to Fig. 2.

V. THE Sn ANALOG

Isovalent and isostructural SnPd3, with a 1.5% smallerlattice constant than PbPd3, was also investigated. Asexpected, the band structure of SnPd3, presented in Fig.10(a), is very similar to that of PbPd3, using GGA only.However, in SnPd3 effects of SOC are not as substantialas in PbPd3, since the strength of SOC in Sn is weakerthan in Pb. The GGA+SOC band structure and thecorresponding DOS are displayed in Figs. 10(b) and (c).Comparing with that of PbPd3, there are two distinctionsworth noting. First, the flat band, which is shifted down-ward by SOC, lies at 10 meV versus 6 meV for PbPd3.Second, the bottom conduction band lies very close to,but above, EF along the Γ − R. These two features re-sult in change in fermiology of SnPd3, shown in Fig. 3(b)beside that of PbPd3. The radius of the pipe-like FS isreduced and irregular rod-shape FSs appears along the111 directions.

VI. CONCLUSION

VII. ACKNOWLEDGMENTS

We acknowledge T. Siegrist for discussions of thephase diagram of PbPd3, and Y.-K. Kim for useful dis-cussion on the topological properties. This researchwas supported by NRF of Korea Grant No. NRF-2016R1A2B4009579 (K.H.A and K.W.L), and by DOEXXXX (W.E.P)

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VIII. SUPPLEMENTAL MATERIAL

Additional details that were mentioned in the main texare provided here, with description in the figure captions.

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FIG. 11: Enlarged band structures of PbPd3 for tetragonally (top) and orthorhombically (bottom) distorted structures. Forcomparison, the notation of the high symmetry points follows that of the cubic case. The X, Y , and Z points are the zoneboundary of (100), (010), and (001), respectively. For the tetragonal case, the C3v symmetry is broken along (111) direction.In the orthrombic case, the C4v symmtery is also broken along (110) direction at kz = π/c plane.

FIG. 12: For the tetragonally distorted structure, plot of the hybrid Wannier charge centers (HWCCs) plot (red, thick lines)across half of the Brillouin zone in the (a) kz = 0 and (b) π/c plane, showing an even number of crossings between the chargecenter and largest gap among two adjacent HWCCs. The orthorhombic case shows similar behavior, so the figure is not repeatedhere. The magnitude of the wave vector ky in the horizontal axis is given in the unit of π/a.

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FIG. 13: (Color online) Enlarged M -centered (001) surface spectral functions for (top) Pd2 and (bottom) Pb-Pd terminationsin GGA. The strong yellowish lines denote surface states. Panels (a) and (d) indicate states at the Fermi energy EF ; (b) and (e)indicate states at 0.18 eV where the triple nodal points (TNPs) appear. Panels (c) and (f) selects only the surface contributionof (b) and (e) respectively. The R-centered spheroid is connected to four protrusions by Fermi arcs. In panels (c) and (f), thegreen dots denote TNPs.

FIG. 14: (Color online) Enlarged M -centered (001) surface spectral functions for (top) Pd2 and (bottom) Pb-Pd terminationsin GGA+SOC. Panels (a) and (d) are at EF . Panels (b) and (e) are at 0.14 eV, crossing the Dirac point along the R−M line.Panels (c) and (f) are at 0.22 eV, crossing the Dirac point along the Γ−R line.

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