Chiral Anomaly Ky Kx

Home , Duncan Haldane, Konstantin Novoselov

Topological Materials

Claudia Felser the revolution • Non-trivial Phenomena captured by Single-Particle Picture → predictive design of novel materials and functionalities • Novel Transport Phenomena determined by Global Properties • quantized transport from topological protection • giant responses …

• Vision:

• Quantum effects at room temperature, spintronics

• Catalysis, energy conversion

• … How does a materials scientist find topological materials? outline

• Introduction • Topological insulators • Weyl semimetals – broken symmetry • Magnetic Weyl semimetals • New Fermions • Outlook – axion insultors and beyond outline

• Introduction • Topological insulators • Weyl semimetals – broken symmetry • Magnetic Weyl semimetals • New Fermions • Outlook – axion insultors and beyond the predictions

predictive materials Topological quantum chemistry design Bradlyn, et al., Nature 2017 1807.10271, 1807.08756 © Nature new concepts control/ for topological Lian, et al., PNAS 2018 synthesis functionality classes: magnetic correlated higher order …

verification of emergent electronic phenomena structure

Qi et al. Nat. Com. (2016) Liu et al. Nat. Mat. (2016) the materials

•explorative search for new materials & predictive design •high quality single crystal growth •epitaxial growth of ultrathin films and heterostructures •2D materials and micro/nano structures

Co2MnGa

MgO the materials

•explorative search for new materials & predictive design •high quality single crystal growth •epitaxial growth of ultrathin films and heterostructures •2D materials and micro/nano structures

Co2MnGa

MgO the measurements

Giant Responses family of quantum Hall effects

1985 Klaus von Klitzing 1998 S Oh Science 340 (2013) 153 Horst Ludwig Störmer and Daniel Tsui 2016 2010 David Thouless, Duncan Haldane und Michael Kosterlitz Andre Geim and Konstantin Novoselov topology in the electronic structure topology in the electronic structure

Magic electron numbers

Hückel: Möbius: 4n+2 aromatic 4n aromatic 4n antiaromatic 4n+2 antiaromatic topology in the electronic structure outline

• Introduction • Topological insulators • Weyl semimetals – broken symmetry • Magnetic Weyl semimetals • New Fermions • Outlook – axion insultors and beyond topological insulators – quantum spin Hall

Heavy insulating elements?

Strained a-Sn and Bi-bilayer

First prediction in graphene by Kane 2 λSOC~ Z for valence shells Kane and Mele, PRL 95, 146802 (2005) topological insulators – quantum spin Hall

- +

Inert pair effect

Kane and Mele, PRL 95, 146802 (2005) Bernevig, et al., Science 314, 1757 (2006) Bernevig, S.C. Zhang, PRL 96, 106802 (2006) König, et al. Science 318, 766 (2007) CO08CH 11-Yan-Felser ARI 31 December 2016 13:25

topological insulators – quantum spin Hall

. SOC

y -

l n

+ o

s Weyl points

s r

g WSM

s Band inversion DSM

Inert pair effect .

v 7

e b

r /

l Dirac point

/ n

Kane and Mele, PRL 95, 146802 (2005) 3

Bernevig, et al., Science 314, 1757 (2006) a

n C = 0 C = 1 o

Bernevig, S.C. Zhang, PRL 96, 106802 (2006) m

o z

König, et al. Science 318, 766 (2007) r

f c d

v Hole .

i Electron

u P

Type-I WSM Type-II WSM

a n

n Figure 1

s o

n T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling

o 9

0 (SOC) opens a full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

. v

B (b)In aWSM /DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some

R W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

u b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

d e

A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

v symmetry or time-reversal symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero

o r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM.

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e c

c hole pockets in the FS.

which iscalled aFermi arc. T he Fermi arc isapparently different from the FSof aTI, an ordinary insulator, or a normal metal, which is commonly a closed loop. T herefore, the Fermi arc offers strong evidence for identifying a W SM by a surface-sensitive technique such as angle-resolved photoemission spectroscopy (ARPES). If T RS exists in a W SM, at least two pairs of W eyl points may exist, where T RS transforms one pair to the other by reversing the chirality. T he Fermi arc still appears, as we discussin thisreview. H owever, the AH E diminishesbecause the Berry phases contributed from two W eyl pairscancel each other. Instead, an intrinsic spin H all effect arises(34) that can be considered as the spin-dependent Berry phase and remains invariant under the T RS.

www.annualreviews.org • Topological Materials 11.3

Changes may still occur before final publication online and in print topological insulators

Starting with Bismuth Bi-Sb Legierungen Bi2Se3 und verwandte Strukturen

Moore and Balents, PRB 75, 121306(R) (2007) Fu and Kane, PRB 76, 045302 (2007) Murakami, New J. Phys. 9, 356 (2007) Hsieh, et al., Science 323, 919 (2009) Xia, et al., Nature Phys. 5, 398 (2009); Zhang, et al., Nature Phys. 5, 438 (2009) CO08CH 11-Yan-Felser ARI 31 December 2016 13:25

topological insulators a

. SOC

s Weyl points

s r

g WSM

s Band inversion DSM

v 7

e b

r /

l Dirac point

n C = 0 C = 1

f c d

v Hole .

i Electron

u P

Type-I WSM Type-II WSM

a n

n Figure 1

s o

n T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling

o 9

0 (SOC) opens a full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

. v

B (b)In aWSM /DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some

R W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

u b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

d e

A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

v symmetry or time-reversal symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero

o r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM.

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e c

c hole pockets in the FS.

A which iscalled aFermi arc. T he Fermi arc isapparently different from the FSof aTI, an ordinary insulator, or a normal metal, which is commonly a closed loop. T herefore, the Fermi arc offers strong evidence for identifying a W SM by a surface-sensitive technique such as angle-resolved photoemission spectroscopy (ARPES). If T RS exists in a W SM, at least two pairs of W eyl points may exist, where T RS transforms one pair to the other by reversing the chirality. T he Fermi arc still appears, as we discussin thisreview. H owever, the AH E diminishesbecause the Berry phases contributed from two W eyl pairscancel each other. Instead, an intrinsic spin H all effect arises(34) that can be considered as the spin-dependent Berry phase and remains invariant under the T RS.

www.annualreviews.org • Topological Materials 11.3

Changes may still occur before final publication online and in print chemistry

Claudia Felser and Xiao-Liang Qi , Guest Editors, MRS Bull. 39 (2014) 843.

Tl+1 Sn2+ Bi+3 inert pair effect predicting topological insulators

Hg Te

Bernevig et al., Science 314, 1757 (2006), König et al., Science 318 (2007) 766. 1, 2, 3, …

ZnS Heusler XYZ C1b

Zr Ni Sn

4 + 10 + 4 = 18

2s +6p electrons 2s +6p+10d electrons Ga As

33 ++ 55 == 88

Li Mg As

1 + 2 + 5 = 8

Graf, Felser, Parkin, IEEE TRANSACTIONS ON MAGNETICS 47 (2011) 367 Graf, Felser, Parkin, Progress in Solid State Chemistry Chemistry 39 (2011) 1 predicting topological insulators

ScNiSb CO08CH 11-Yan-Felser ARI 31 December 2016 13:25

LaPtBi

Chadov, et al., Nature Mat. 9 (2010) 541 Lin et al., Nature Mat. 9 (2010) 546

. SOC

s Weyl points

s r

g WSM

s Band inversion DSM

v 7

e b

r /

l Dirac point

n C = 0 C = 1

f c d

v Hole .

i Electron

u P

Type-I WSM Type-II WSM

a n

n Figure 1

s o

n T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling

o 9

0 (SOC) opens a full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

. v

B (b)In aWSM /DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some

R W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

u b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

d e

A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

v symmetry or time-reversal symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero

o r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM.

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e c

c hole pockets in the FS.

www.annualreviews.org • Topological Materials 11.3

Changes may still occur before final publication online and in print unconventional superconductors

1.6 LaPtBi

1.2 2.0 1.5

0.8

1.0

0.4 0.5 cm)

 0.0 0.0 0 1 2 3 4 5 32

Resistivity (m Resistivity 20

16 15 YPtBi

10 8 5 0 0 1 2 3 4 5 YPdBi 0 0 50 100 150 200 250 300 T (K)

YPtBi distorted structure to property

HgTe

LaPtBi a Ag2Te

b Ag2Te AuTlS2 Li2AgSb distorted structure to property

HgTe From 2D to 3D

a Ag2Te LaPtBi

AuTlS2

b Ag2Te Li2AgSb

Nature Mat. 9 (2010) 541 Nature Mat. 9 (2010) 546 arXiv:1004.0999v1 PRL 106, 016402 (2011) PRL 106, 156808 (2011) arXiv:1104.0335v1 Müchler, Zhang, Chadov, Yan, Kübler, Zhang, Felser, Angewandte Chemie 2012 distorted structure to property

HgTe

Graphite LaPtBi a Ag2Te

b Ag2Te AuTlS2 Li2AgSb KHgSb Graphite

j=3/2  M K 

j=1/2 LiAuSn

Hg-s

KHgSb

Yan, Müchler, Felser, Phys. Rev. Lett. 109 (2012) 116406 Zhang, Chadov, Müchler, Yan, Qi, Kübler, Zhang, Felser, Phys. Rev. Lett. 106 (2011) 156402 honeycomb from sp3 to sp2

Band inversion is found in the heavier compounds No surface state? Why ? Interaction between the two layers in the unit cell and two Dirac Cones

Zhang, Chadov, Müchler, Yan, Qi, Kübler, Zhang, Felser, Phys. Rev. Lett. 106 (2011) 156402 arXiv:1010.2195v1 honeycomb from sp3 to sp2

Yan, Müchler, Felser, Phys. Rev. Lett. 109 (2012) 116406 Weyl semimetals

Breaking inversion symmetry Dirac semimetals

Bohm-Jung Yang and Naoto Nagaosa, arXiv:1404.0754 CO08CH 11-Yan-Felser ARI 31 December 2016 13:25

Weyl semimetals

. SOC

s Weyl points

s r

g WSM

s Band inversion DSM

v 7

e b

r /

l Dirac point

n C = 0 C = 1

o z

Binghair Yan and Claudia Felser, Annual Review in Condensed Matter 8 (2017) 337

f c d

v Hole .

i Electron

u P

Type-I WSM Type-II WSM

a n

n Figure 1

s o

n T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling

o 9

0 (SOC) opens a full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

. v

B (b)In aWSM /DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some

R W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

u b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

d e

A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

v symmetry or time-reversal symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero

o r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM.

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e c

c hole pockets in the FS.

www.annualreviews.org • Topological Materials 11.3

Changes may still occur before final publication online and in print Weyl semimetals

▪ Breaking symmetry ▪ Inversion symmetry (Structural distortion) E with

k ▪ Breaking time reversal symmetry E without ▪ Magnetic field

Dirac points are at high symmetry points k Weyl points are not at high symmetry points the materials

RPtBi, RPdBi, RPtSb Bi Se , Bi Te Sb Te Bi Te Se, Topological Insulators 2 3 2 3 , 2 3 2 2 Sb2Te 2Se TlBiTe2 TlBiSe2, TlBiSSe2

Cd As , Na Bi, ZrSiS, HfSiS Topological 3 2 3 LaBi, LaSb, PtSe , PtTe Dirac semimetals 2 2 MA3 (M=V, Nb, Ta; A=Al, Ga, In), Topological CuMnAs, HfTe5 materials NbP,TaP,NbAs,TaAs,WTe2, Td- Topological MoTe2 Weyl semimetals Co2TiX (X=Si, Ge, Sn) Mn3Ge, Mn3Sn, Mn3Ir, Mn3Ga, RPtBi, WP2, MoP2

Beyond-Topological MoP, PdSb2, La4Bi3, Mg3Ru2, Dirac and Weyl Ta 3Sb, Nb3Bi, CoSi, RhSi, AlPt, PdGa high quality materials

•large mean free path, high mobility 107 •large magneto resistance, quantum oscillations NbP LaSb PtBi2 WP 106 2 •low defect density WTe2 MoP2 5 high conductivity 10 PtSn Cd As • NbSb 4 3 2 TaAs 2 104 18 3 IrO MoP 10 2 Cu-4N 15 Cu 2 K

cm) 10 Magnetoresistance(%)

  12

n (

1 xx a 10  5 6 7 8 9 10 9 10 10 10 10 10 10 MoP Conductivity (-1cm-1) 6 0 5 10 15 20 25 T (K) Hydrodynamic flow of electrons

C. Shekhar et al. Nat. Phys. 2015, N. Kumar et al. Nat. Commun 2017, J. Gooth et al. Nat. Commun. 2018, Gooth Nature 2017 CO08CH 11-Yan-Felser ARI 31 December 2016 13:25 Weyl semimetals

3D topological Weyl in transportabmeasurement: B

E 1. Fermi arc

2. Intrinsic anomalous Hall effect v E v 3. Planar Hall effect F 4. Chiral anomaly ky kx

5. Axial. gravitational anomaly

y + –

e + Fermi arc –

l a

n Figure 2

g r

e Schematics of Fermi arcs and the chiral anomaly effect. (a) Existence of Fermi arcs in the Fermi surface of

r s

o the surface band structure. A pair of bulk W eyl cones exists as a pair of Fermi pockets at a Fermi energy

. EF ̸= 0oraspointsat EF = 0, where the pink (chartreuse) color represents the + (− )chirality. AFermi arc

1 r

/ (thick line) appears on the top or bottom surface to tangentially connect such a pair of Fermi pockets. (b)The

2 u

/ chiral anomaly can be simply understood with the zeroth Landau level of a W eyl semimetal in the quantum

a limit. T he zeroth Landau levels from the + and − chiral W eyl conesexhibit opposite velocities due to

o m

different chirality. Applied electric ﬁeld leads to an imbalance of electron densities in the left-hand and

f i

right-hand valleys, breaking the number conservation of electrons at a given chirality. T his result, however,

a d

AA Burkov,e L Balents, PRL 107 12720 (2012)

M isnot limited to the quantum limit and was recently proposed to induce an anomalous DC current that is

AA Burkov,f arXiv:1704.05467v2 o

o quadratic in the ﬁeld strength in the semiclassical limit.

l y

AA Burkov,n J. Phys.: Condens. Matter 27 (2015) 113201

v An important consequence of the 3D W eyl band structure is that W SMs display the chiral

. 7

U anomaly effect (see Figure 2). In high-energy physics, the particle number of W eyl fermions

r 2

e for a given chirality is not conserved quantum mechanically in the presence of nonorthogonal

n y

e magnetic (B) and electric (E) ﬁelds (i.e., E · B isnonzero), inducing a phenomenon known as the

r Adler–Bell–Jackiw anomaly or chiral anomaly (35, 36). In W SMs, the chiral anomaly ispredicted

e a

n to lead to anegative magnetoresistance (M R) owing to the chiral zero modesof the Landau levels

h of the 3D W eyl cones and the suppressed backscattering of electrons of opposite chirality (37,

- 38). T he negative MRisexpected to exhibit the largest amplitude when B E, because the B E

d ∥ ⊥

7 o

9 component contributesapositiveMRduetotheL orentz force. Inaddition tothenegativeM R, the

. v

B chiral anomaly isalso predicted to induceexotic nonlocal transport and optical properties(39–41).

. M any material candidates have been predicted as W SMs, e.g., the pyrochlore iridates

d Y2Ir2O7 (11), H gCr2Se4 (30), and Hg1− x− yCdxMnyTe(42). H owever, these candidates have not

A d

i been experimentally realized thus far. In early 2015, four W SM materials—TaAs, T aP, N bAs,

v o

r and N bP—were discovered through calculations (19, 20) and the observation of Fermi arcs us-

s ing ARPES (21–23), realizing W eyl fermions for the ﬁrst time [also in photonic crystals (43)].

e c

c M eanwhile, many efforts have been devoted to their magnetotransport properties (44–47). T hese A W SM compounds preserve T RS but break crystal inversion. In addition, the DSMs were found

to exist in Na3Bi and Cd3As2 by ARPES (48–51). W SMscan be classiﬁed into type I,which respects Lorentz symmetry, and type II,which does not (see Figure 1) (52). T he T aAs family of W SMs exhibits ideal W eyl cones in the bulk band structure and belongsto thetype-I class, i.e., the FSshrinksto apoint at theW eyl point. M orere- cently, type-II W SMshave been proposed to exist in the layered transition-metal dichalcogenides

W T e2 (52) and its sister compound M oT e2(53). H ere, the W eyl cone exhibits strong tilting so that the W eyl point is the contact point between an electron pocket and a hole pocket in the FS. T ype-II W SMs are expected to show very different properties from type-I W SMs, such as

11.4 Yan · Felser

Changes may still occur before final publication online and in print CO08CH 11-Yan-Felser ARI 31 December 2016 13:25

SOC

s Weyl points

s r

g WSM

s Band inversion DSM

v 7

e b

r /

l Dirac point

/ n

3 type I or II

n C = 0 C = 1

f c d

v Hole .

i Electron

u P

Type-I WSM Type-II WSM

G r

www.advancedsciencenews.com www.advmat.de

a n

n Figure 1

o n

T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of O

J is quite different. As mentioned earlier, 1T-TaS2 shows almost

Graphenee

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling n

7 no HER activity whereas 1T¢-MoTe2 shows a very high activity.

o 9

0 (SOC) opensa full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

6 Si nce few-layers 1T¢-MoTe rather exhibits topological features .

2 M v

B (b)In aWSM/DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some I

e in its band structure, this has encouraged us to consider the pos- R

U W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

. y

u sible role of topological effects. We consider below the recently b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

n d

e discovered Weyl semimetals, NbAs, TaAs, NbP, and TaP. A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

v symmetry or time-reversal symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero In a Weyl semimetal, the conduction and valence bands cross o

A r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM .

each other linearly through nodes (Figure 3a), called the Weyl

T s

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e points, near the Fermi energy. As a 3D analogue of graphene, c

c hole pockets in the FS.

A topological Weyl semimetals (TWSs) are expected to exhibit very

high mobility in their charge transport.[11] Similar to TIs, TWSs N which iscalled aFermi arc. T heFermi arc isapparently different from the FSof aTI, an ordinary [25] A. K. Geim, A. H. MacDonald Physics Today, 08.(2007), 35-41 Shekhar Chandra, et al. , Nature Physics 11 (2015)also 645present robust metallic surface states that are stable insulator, or a normal metal, which iscommonly a closed loop. T herefore, the Fermi arc offers against defects, impurities, and other surface modifications. strong evidence for identifying a W SM by a surface-sensitive technique such as angle-resolved Analogous to the role of graphene, in the MoS2 catalyzed HER, photoemission spectroscopy (ARPES). If T RS existsin a W SM, at least two pairsof W eyl points we believe that the highly mobile TWS bulk states help electrons may exist, where T RS transforms one pair to the other by reversing the chirality. T he Fermi arc diffuse freely and quickly. Furthermore, the topological surface still appears, as we discussin thisreview. H owever, the AH E diminishesbecause the Berry phases states may cause the surface to act as stable active planes for contributed from two W eyl pairscancel each other. Instead, an intrinsic spin H all effect arises(34) catalysis. The first family of TWSs that was experimentally dis- that can be considered as the spin-dependent Berry phase and remains invariant under the T RS. covered, from direct observations of their topological surface states, was the transition metal monopnictide: NbP, TaP, NbAs, [26–30] www.annualreviews.org • Topological Materials 11.3 and TaAs. These materials are semimetals wherein Weyl points are located near the Fermi level with a total of 12 pairs of Weyl nodes in the first Brillouin zone. For this reason, we have investigated the HER activity in these TWS compounds. The HER activities of NbP, TaP, NbAs, and TaAs were studied over a period of 6 h. Our studies show that all four TWSs are highly HER active (Figure 3c) and NbP, being the Changes may still occur before final publication online and in print lightest among all, performs the best as an HER catalyst with the highest value of H 2 evolved per gram of the catalyst (3520 mmol g-1). The compounds can undergo many cycles of HER without activity fading as can be seen in Figure 4b, where we show three cycles of HER in NbP with a comparable catalytic performance each time. Chemical analysis shows no observable changes in chemical composition of our catalysts (Figure S9, Supporting Information) af ter several HER cycles. We show the activity and turnover frequency (TOF: the number of moles of H 2 evolved per mole of catalyst used) as histograms for all four compounds in Figure 3d. In general, phosphides are better HER catalysts than arsenides. We note that all 4 compounds are WSMs with well-defined and distinct Weyl points and each has very high mobilities from the linearly dispersed bands at the Weyl points, which accounts for their high catalytic Figure 3. Electronic band structure of topological Weyl semimetals and activities. We therefore expect that the catalytic HER properties their HER activity. a) Schematic band structure of the transition metal within this series will be determined by the chemical bonding monopnictide TWS family, revealing semimetallic character. Weyl nodes of hydrogen at the surface, which is reflected in the value of of opposite chiralities are marked with blue and red dots. b) Comparison of hydrogen evolution activity of various TWSs (NbP, TaP, NbAs, and DGH*. Indeed, we find that their HER activity is correlated with TaAs) powdered single crystals with an intermediate dye addition. c) His- the DGH* values for these compounds. NbP has the lowest togram of hydrogen evolution rate and TOF, shown on left and right axes, DGH* among all these compounds followed by TaP, TaAs, and respectively, for all four compounds. NbAs, and TOF also follows a similar trend. Having investigated the thermodynamic aspects of the the three compounds where we can clearly see that the topo- catalysts we now focus on the role of kinetics. As we know logically nontrivial 1T¢-MoTe2 is the most active catalyst. that the reduction of water occurs at the surface of the cata- The Gibb’s free energy (DGH*) of adsorption of hydrogen at lyst, increasing the surface area of the catalyst should result the catalyst surface is very often used to predict the activity of an in increased activity of the catalyst. For this we have selected HER catalyst. The closer this value is to zero the better is the per- NbP as an example and compared the activity in single crystals formance. The DGH* (on abscissa) and the activity (on ordinate) crushed into powder (few mm in size, Figure S9, Supporting hence make a so-called volcano diagram (Figure 2c). Notwith- Information) and polycrystalline material (150–300 nm in standing that both 1T-TaS2 and 1T¢-MoTe2 are metallic with com- size) obtained by solid state reaction. We encounter a twofold parable DGH* values, the HER activity of these two compounds increase in the activity of polycrystals as compared to the single

NbP, NbAs, TaP, TaAs www.advancedsciencenews.com www.advmat.de

is quite different. As mentioned earlier, 1T-TaS2 shows almost O

no HER activity whereas 1T¢-MoTe2 shows a very high activity. M Si nce few-layers 1T¢-MoTe rather exhibits topological features

2 M in its band structure, this has encouraged us to consider the possible role of topological effects. We consider below the recently U

N discovered Weyl semimetals, NbAs, TaAs, NbP, and TaP.

In a Weyl semimetal, the conduction and valence bands cross C each other linearly through nodes (Figure 3a), called the Weyl A

points, near the Fermi energy. As a 3D analogue of graphene, I topological Weyl semimetals (TWSs) are expected to exhibit very O high mobility in their charge transport.[11] Similar to TIs, TWSs N also present robust metallic surface states[25] that are stable against defects, impurities, and other surface modifications. Analogous to the role of graphene, in the MoS2 catalyzed HER, we believe that the highly mobile TWS bulk states help electrons diffuse freely and quickly. Furthermore, the topological surface states may cause the surface to act as stable active planes for catalysis. The first family of TWSs that was experimentally discovered, from direct observations of their topological surface states, was the transition metal monopnictide: NbP, TaP, NbAs, and TaAs.[26–30] These materials are semimetals wherein Weyl points are located near the Fermi level with a total of 12 pairs of Shekhar, et al. , Nature Physics 11 (2015) 645, Weng, et al. Phys. Rev. X 5, 11029 (2015) Weyl nodes in the first Brillouin zone. For this reason, we have Arnold, et al., Nature Communication 7 (2016) 11615 Huang et al. preprint arXiv:1501.00755 investigated the HER activity in these TWS compounds. The HER activities of NbP, TaP, NbAs, and TaAs were studied over a period of 6 h. Our studies show that all four TWSs are highly HER active (Figure 3c) and NbP, being the lightest among all, performs the best as an HER catalyst with the highest value of H 2 evolved per gram of the catalyst (3520 mmol g-1). The compounds can undergo many cycles of HER without activity fading as can be seen in Figure 4b, where we show three cycles of HER in NbP with a comparable catalytic performance each time. Chemical analysis shows no observable changes in chemical composition of our catalysts (Figure S9, Supporting Information) af ter several HER cycles. We show the activity and turnover frequency (TOF: the number of moles of H 2 evolved per mole of catalyst used) as histograms for all four compounds in Figure 3d. In general, phosphides are better HER catalysts than arsenides. We note that all 4 compounds are WSMs with well-defined and distinct Weyl points and each has very high mobilities from the linearly dispersed bands at the Weyl points, which accounts for their high catalytic Figure 3. Electronic band structure of topological Weyl semimetals and activities. We therefore expect that the catalytic HER properties their HER activity. a) Schematic band structure of the transition metal within this series will be determined by the chemical bonding monopnictide TWS family, revealing semimetallic character. Weyl nodes of hydrogen at the surface, which is reflected in the value of of opposite chiralities are marked with blue and red dots. b) Comparison of hydrogen evolution activity of various TWSs (NbP, TaP, NbAs, and DGH*. Indeed, we find that their HER activity is correlated with TaAs) powdered single crystals with an intermediate dye addition. c) His- the DGH* values for these compounds. NbP has the lowest togram of hydrogen evolution rate and TOF, shown on left and right axes, DGH* among all these compounds followed by TaP, TaAs, and respectively, for all four compounds. NbAs, and TOF also follows a similar trend. Having investigated the thermodynamic aspects of the the three compounds where we can clearly see that the topo- catalysts we now focus on the role of kinetics. As we know logically nontrivial 1T¢-MoTe2 is the most active catalyst. that the reduction of water occurs at the surface of the cata- The Gibb’s free energy (DGH*) of adsorption of hydrogen at lyst, increasing the surface area of the catalyst should result the catalyst surface is very often used to predict the activity of an in increased activity of the catalyst. For this we have selected HER catalyst. The closer this value is to zero the better is the per- NbP as an example and compared the activity in single crystals formance. The DGH* (on abscissa) and the activity (on ordinate) crushed into powder (few mm in size, Figure S9, Supporting hence make a so-called volcano diagram (Figure 2c). Notwith- Information) and polycrystalline material (150–300 nm in standing that both 1T-TaS2 and 1T¢-MoTe2 are metallic with com- size) obtained by solid state reaction. We encounter a twofold parable DGH* values, the HER activity of these two compounds increase in the activity of polycrystals as compared to the single

Increasing spin orbit coupling increases – heavier elements Distance between the Weyl points increases

NbAs TaP

Liu et al., Nature Mat. 15 (2016) 27 Yang, et al. Nature Phys. 11 (2015) 728 chiral anomaly

• Chiral anomaly is the anomalous non-conservation of a chiral current. • A sealed box with equal number of positive and negative charged particles, its found when it is opened to have more positive than negative particles, or vice-versa. • Events are expected to be prohibited according to classical conservation laws, must be ways they can be broken, because the observable universe contains more matter than antimatter Wikipedia Vacuum Dirac state elementary semimetal particles 'electrons'

E v = c E v = v ≈ 0.001 c ky ky F kx kx 'holes' anti-particles

vacuum dispersion band structure

S. L. Adler, Phys. Rev. 177, 2426 (1969) J. S. Bell and R. Jackiw, Nuovo Cim. A60, 47 (1969) AA Zyuzin, AA Burkov - Physical Review B (2012) chiral anomaly

Ga-doping relocate the Fermi energy in NbP close to the W2 Weyl points Therefore we observe a negative MR as a signature of the chiral anomaly the, NMR survives up to room temperature Niemann, et al. Scientific Reports 7 (2017) 43394 doi:10.1038/srep4339 preprint arXiv:1610.01413 axial-gravitational anomaly

Experimental signatures for the mixed axial-gravitational anomaly in Weyl semimetals

• In solid state physics, mixed axial-gravitational anomaly can be identified by a positive magneto-thermoelectric conductance (PMTG) for DT ll B.

• Low fields: quadratic

B • High fields: deminishes

• DT ll B dictates sensitivity on alignement of B and DT. GT

[Lucas, et al. Proceedings of the National Academy of Sciences 113, 9463 (2016)] gravitational anomaly

• Landsteiner K., et al. Gravitational anomaly and transport phenomena. Phys. Rev. Lett. 107, 021601 (2011). URL • Jensen, et al. Thermodynamics, gravitational anomalies and cones. Journal of High Energy Physics 2013, 88 (2013). • Lucas, A., Davison, R. A. & Sachdev, S. Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in weyl semimetals. PNAS 113, 9463–9468 (2016).

A positive longitudinal magneto-thermoelectric conductance (PMTC) in the Weyl semimetal NbP for collinear temperature gradients and magnetic fields that vanishes in the ultra quantum limit.

Gooth et al., Nature 547 (2017) 324 arXiv:1703.10682 universe – in a crystal particles and quasi particles

Mattuck, “A Guide to Feynman Diagrams in the Many-Body Problem” (1967) GdPtBi – an ideal Weyl

C. Shekhar et al., PNAS 2018, arXiv:1604.01641, (2016), Kumar PRB 2018, Kaustuv Nat. Mat. Rev. (2019). M. Hirschberger et al.,. Nature Mat. Online arXiv:1602.07219, (2016). Weyl semimetals

Breaking time reversal symmetry CO08CH 11-Yan-Felser ARI 31 December 2016 13:25 Weyl semimetals

Breaking symmetry a ▪ Inversion symmetry (Strain) Breaking time reversal symmetry TI ▪ Magnetic field

. SOC

s Weyl points

o s

r ▪

g WSM Dirac points at high symmetry

r s Band inversion DSM o ▪

w Weyl points at low symmetry

v 7

e b

1 ▪

r All crossings in ferromagnets: /

l Dirac point

3 n

0 Weyl points

n C = 0 C = 1

a d

e Binghai Yan and Claudia Felser, Annual Review in Condensed Matter 8 (2017) 337

f c d

v Hole .

i Electron

u P

Type-I WSM Type-II WSM

a n

n Figure 1

s o

n T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling

o 9

0 (SOC) opens a full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

. v

B (b)In aWSM /DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some

R W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

u b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

d e

A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

v symmetry or time-reversal symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero

o r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM.

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e c

c hole pockets in the FS.

www.annualreviews.org • Topological Materials 11.3

Changes may still occur before final publication online and in print Berry curvature design

Berry curvature design Nernst effect • giant spin Hall • giant anomalous Hall • giant topological Hall • giant anomalous Nernst

J. Noky et al., Phys. Rev. B 98, 241106(R) (2018), arXiv:1807.07843, arXiv:1806.06753 Heusler, Weyl and Berry

cm) 2



-7 0 (10

yx -2 

-4

-10 -8 -6 -4 -2 0 2 4 6 8 10 H (T)

For the planar cases the AHC is connected with Weyl points in the energy- band structure.

Chen, Niu, and MacDonald, Phys. Rev. Lett., 112 (2014) 017205. Kübler and Felser EPL 108 (2014) 67001 Heusler, Weyl and Berry

Mn3Ge Mn3Sn

The anomalous Hall conductivity in an antiferromagnetic metal is zero Nayak et al. arXiv:1511.03128, Science Advances 2 (2016) e1501870 Kiyohara, Nakatsuji, preprint: arXiv:1511.04619 Nakatsuji, Kiyohara, & Higo, Nature, doi:10.1038/nature15723 Fermiarcs and chiral anomaly in the Weyl AFM

Hao Yang, et al. New J. Phys. 19 (2017) 015008, Nernst effect in Mn3Sn

Magnetization dependence of the spontaneous a) Hall resistivity, ρxz); (b) Hall conductivity, σxz, extracted from ρxz, ρxx, and ρzz; (c) Nernst signal, Sxz; (d) Transverse Nernst effect for ferromagnetic metals and Mn3Sn thermoelectric conductivity, αxz, Kagome lattice

Mn,Fe,Co

Fe3Sn2

Nature, 2018, doi:10.1038/nature25987 Kang et al., preprint arXiv:1906.02167 Kagome lattice

Looking for Weyl fermions on a ferromagnetic Kagomé lattice with out of plane magnetisation.

Fe3Sn2 Co3Sn2S2

Nature, 2018, doi:10.1038/nature25987 Enke Liu, et al. Nature Physics 14 (2018) 1125 , preprint arXiv:1712.06722 Weyl and Berry

Weyl points are close to EF Co3Sn2S2 Hard magnetic behaviour

transport: chiral anomaly

D. F. Liu, et al., Science (2019) under review Liu, et al. Nature Physics 14 (2018) 1125 , preprint arXiv:1712.06722 Weyl and Berry

STM and ARPES confirms Weyl and Fermiarcs

D. F. Liu, et al., Science (2019) under review Morali et al., Science accepted , preprint arXiv:1903.00509 Weyl and Berry

ARPES confirms Weyl and Fermiarcs

D. F. Liu, et al., Science (2019) under review Weyl and Berry

Morali et al., Science accepted , preprint arXiv:1903.00509 new transport properties

Berry curvature design Co3Sn2S2 • giant anomalous Hall Mn3Sn Giant Hall Angle 20% • giant anomalous Nernst

Liu, et al. Nature Physics 14 (2018) 1125 , Guin, et al. Adv. Mater. 2019, 1806622, arXiv:1712.06722, arXiv:1807.07843, arXiv:1806.06753 new transport properties

Berry curvature design Co MnGa • giant anomalous Hall 2 Co3Sn2S2 • giant anomalous Nernst Mn3Sn

3 DT || ab plane

Magnetized with positive field

)

-1 V K V

 0 Zero field Nernst signal region

(

xy S Magnetized with negative field

-3 0 50 100 150 T (K) Guin, et al. Adv. Mater. 2019, 1806622, arXiv:1712.06722, arXiv:1807.07843, arXiv:1806.06753 new transport properties

Berry curvature design Co MnGa • giant anomalous Hall 2 Co3Sn2S2 • giant anomalous Nernst Mn3Sn Co Sn S (present work) 35 3 2 2

3 DT || ab plane M 0 

Magnetized with positive field to

)



-1

A xy

V K V 30

S 

0 Zero field Nernst signal region MnGa



(

2 O

Magnetized with negative field 2

4 Mo

5 -Mn

-FePt

-FePd

-MnGa

Ratio of Ratio

Co/Ni films Co/Ni

Fe/Pt multilayer, N=9 multilayer, Fe/Pt MnGe (20 < B K, T) 14 -3 MnGe K, (100 > B T) 5 0 50 100 150 0 T (K) Samples Guin, et al. Adv. Mater. 2019, 1806622, arXiv:1712.06722, arXiv:1807.07843, arXiv:1806.06753 quantum devices

1.0

0.5

) 2 0.0 R

h/e xx (

R Ryx -0.5

-1.0 -1.0 -0.5 0.0 0.5 1.0 • Towards QAHE in MBE grown thin films. B (T) • The anomalous Hall resistance is more than Cr-doped (Bi,Sb) Te 0.75 h/e2, comparable to the QAHE sample 2 3 in previous reports. quantum devices

Bulk WSM Film QAH

A σxy 1 • Towards QAHE in MBE grown thin films. kz MMkkk=++ ()222 • Magnetic Weyl for QAH effect in 2D 01xyz

Qiunan Xu, Enke Liu, Wujun Shi, Lukas Muechler, Claudia Felser, Yan Sun, preprint arXiv:1712.08915 the Heusler Family

Diamond ZnS Heusler XYZ C1b X2YZ L21

Graf, Felser, Parkin, IEEE TRANSACTIONS ON MAGNETICS 47 (2011) 367 Graf, Felser, Parkin, Progress in Solid State Chemistry Chemistry 39 (2011) 1

Heusler, Weyl and Berry

the tuning parameter - topology is more than S symmetries - insulators, semiconductors, metals, topological insulators, Weyl and Dirac, spin gapless semiconductors - spin – ferro, ferri, antiferro, compensated ferri, non collinear - spin – in plane, out of plane - doping and substitution to adjust the chemical potential, rigid band model - from isotropic to anisotropic: cubic, tetragonal and hexagonal distortion - phase transitions

Manna et al., Nature Materials Review, 2018, Nayak, Nature 2017, Nayak, Science Advances 2017 Heusler, Weyl and Berry

Half-Heusler, F4ത3푚 (no. 216)

Regular-Heusler, F푚3ത푚 (no. 225) Inverse-Heusler, F4ത3푚 (no. 216)

Manna et al., Nature Materials Review, 2018, Nayak, Nature 2017, Nayak, Science Advances 2017 From semiconductors to half metals

▪ magic valenceSynthesis electron number: 24 X2YZ ▪ valence electrons = 24 + magnetic moments

EF Co Mn Ga Fe2 V Al 2

2 * 8 + 5 + 3 = 24 2 * 9 + 7 + 3 = 24 + 4

Kübler et al., PRB 28, 1745 (1983) Kandpal et al., J. Phys. D 40 (2007) 1507 de Groot RA, et al. PRL 50 2024 (1983) Balke et al. Solid State Com. 150 (2010) 529 Galanakis et al., PRB 66, 012406 (2002) Kübler et al., Phys. Rev. B 76 (2007) 024414 Tuning exchange

X2YZ Co2TiAl: 29 + 4 + 3 = 25 Ms = 1mB

Co2MnGa: 29 + 7 + 3= 28 Ms = 4mB

Co2FeSi: 29 + 8 + 4 = 30 Ms = 6mB

Heusler and Weyl

. SOC

s Weyl points

Breakingu symmetry Co TiSn

l 2

a n

o ▪ Inversion symmetry (Strain)

s r

g WSM

s Band inversion DSM Co2TiSi: 29 + 4+ 4 = 26 Ms = 2B

F e

Breaking time reversal symmetry

v 7

e b

r / l ▪ Dirac point

9 Magnetic field

n C = 0 C = 1

f c d

v Hole .

i Electron

n y e ▪

t Dirac points at high symmetry

u P

Type-I WSM Type-II WSM

e s

t ▪ t

e Weyl points at low symmetry

a n

n Figure 1

. h ▪

s All crossings in ferromagnets: o

n T he topological insulator (T I) and W eyl semimetal (W SM) or Dirac semimetal (DSM). T he topology of

- d

both a TI and a W SM/DSM originates from similar inverted band structure. (a) T he spin-orbit coupling

7 o 9 Weyl points

0 (SOC) opens a full gap after the band inversion in a TI, giving rise to metallic surface states on the surface.

. v

B (b)In aWSM /DSM,the bulk bandsare gappedbythe SOCin the 3Dmomentumspaceexcept at some

R W

isolating linearly crossing points, namely W eyl points/Dirac points, as a 3D analog of graphene. Due to the

u b

n topology of the bulk bands, T SSs appear on the surface and form exotic Fermi arcs. In a DSM all bandsare

n d

Binghai Yane Claudia Felser, Zhijun Wang, et al., arXiv:1603.00479

A doubly degenerated, whereas in a W SM the degeneracy islifted owing to the breaking of the inversion

d i

Annual Reviewv in Condensedsymmetry Matter 8or (2017time-reversal) 337 symmetry or both. (c)The type-I W SM.The Fermi surface(FS) shrinks tozero Guoqing Chang et al., arXiv:1603.01255

o r

p at the W eyl points when the Fermi energy issufﬁciently close to the W eyl points. (d)The type-II W SM.

s Due to the strong tilting of the W eyl cone, the W eyl point acts as the touching point between electron and

e c

c hole pockets in the FS.

www.annualreviews.org • Topological Materials 11.3

Changes may still occur before final publication online and in print Heusler, Weyl and Berry

Giant AHE in Co2MnAl

 xy =1800 S/cm calc.

 xy  2000 S/cm meas.

푀 2 Kübler, Felser, PRB 85 (2012) 012405 휌푥푦 = a푥푥 + b푥푥 푴. ??? Vidal et al Appl.Phys.Lett. 99 (2011) 132509 Kübler, Felser, EPL 114 (2016) 47005. Heusler, Weyl and Berry

Giant AHE in Co MnAl 2 Co2MnGa

 xy =1800 S/cm calc.

 xy  2000 S/cm meas.

Kübler, Felser, PRB 85 (2012) 012405 Vidal et al., Appl. Phys. Lett. 99 (2011) 132509 Kübler, Felser, EPL 114 (2016) 47005. Berry curvature design

Berry curvature design Nernst effect • giant spin Hall • giant anomalous Hall • giant topological Hall • giant anomalous Nernst

J. Noky et al., Phys. Rev. B 98, 241106(R) (2018), arXiv:1807.07843, arXiv:1806.06753 Nernst effect

Co2MnGa

a b 8 9 Q || [111]

) 6 Q || [111] -1  H || [011] 0  H || [011]

V K 0

6  4

( A

yx 4

S )

) 2 -1

-1 3

100 200 300 K 1 -1

V K T (K) 10

 0 0 Co MnGa (

A m 2

(

yx S -3 340 K 210 K 340 K 210 K a Fe/Pt multilayer 315 K 160 K 315 K 160 K 0 290 K 110 K -4 -6 290 K 110 K ) 10 L1 -MnGa 265 K 60 K 265 K 60 K 0 Fe 240 K -1 DO -Mn Ga Co -9 Heusler, We 240y Kl and Berry Mn Sn 22 2 -8 3 Nd Mo O -2 -1 0 1 2 -2 -1 0 1 2 -1 2 2 7 Co/Ni films V K V 10  H (T)  H (T) 0 0 

( MnGe c 1.0 d 12

T = 300 K 4  Theory ) -2

-1 E = E - 0.08 eV yx 2 F 0 A 10

10 K -1 0.5 S

) 0

 -1

A m A EF = E0

(

MnGe (20 K, B < 14 T)

EF = E0 + 0.08 8 yx -2 -1 a -3 O 4 (eV) E EF = E0+ 0.08 eV 0 0.0 10 3 0 -4

A m A Fe (

0 100 200 300

EF = E0 - 0.08 6

E-E

A yx T (K) a Fe/Pt multilayer, N=4 -0.5 Experiment -4 Fe/Pt multilayer, N=7 4 10 10-4 10-3 10-2 10-1 100 101 -1.0 2 -8 -4 0 4 8 100 200 300 A -1 -1  M (T) ayx (A m K ) T (K) 0

Satya N. Guin, et al., NPG Asia Mater. 11, 16 (2019), arXiv:1806.06753 J. Noky et al., Phys. Rev. B 98, 241106(R) (2018) Sakain et al. Nature Physics 2018 Co2MnGa – ferromagnetic nodal line

Z 4a4b Co 8c Y

Series of ARPES cuts through the candidate line node

Belopolski, et al., Science accepted (2019) preprint arXiv:1712.09992 Heusler, Weyl and Berry

3000 2 K 0H || [100] 50 K Z 2000 100 K

4a 150 K ) 4b Co YZ (Y = IVB or VB; Z = IVA or IIIA) 200 K 2 -1 1000 250 K

Co cm 8c ത 300 K L21 space group 225 (Fm3m) -1 0

Y  (

xy -1000 

-2000

-3000 -10 -8 -6 -4 -2 0 2 4 6 8 10

0H (T) With SOC M 0.12 Anomalous Hall Angle:

D휌푥푦 휌푥푦−푅0휇0퐻 0.11 A = = xx 휌xx 휌푥푥 

Symmetry and electronic structures / xy

depend on the magnetization direction  0.10 = D =  0.09 Without SOC Phys. Rev. Lett. 117, 236401 (2016)  Sci. Rep. 6, 38839 (2016) 0.08 ❑ nodal line is formed in the plane when bands of opposite mirror eigenvalues cross. 0 50 100 150 200 250 300 ❑ Mirror planes are related to each other by the rotations T (K) Manna et al., Phys. Rev. X 8 (2018) 041045, arXiv:1712.10174 no inversion symmetry Half-Heusler, F4ത3푚 (no. 216) topology !

Regular-Heusler, F푚3ത푚 (no. 225) Inverse-Heusler, F4ത3푚 (no. 216) Heusler, Weyl and Berry

playing with symmetry: from Weyl to spingapless semiconductor

3000 2 K

50 K

Co2MnGa ) 2000

100 K -1 1000 150 K

200 K cm 250 K

-1 0

300 K 

( -1000 0H || [100] xy

 -2000 -3000 -9 -6 -3 0 3 6 9 0H (T)

2 K Mn CoGa ) 2000

2 50 K -1 1000 100 K

200 K cm 300 K

-1 -1 0 

(  H || [100]

-1000 0 xy  -2000 -9 -6 -3 0 3 6 9 H (T) Manna et al., Phys. Rev. X 8 (2018) 041045, arXiv:1712.10174 Heusler, Weyl and Berry playing with symmetry: from Weyl to spingapless semiconductor

Co2TiSn

Mn2CoGa

Ouardi, Fecher, Kübler, and Felser, Physical Review Letter 110 (2013) 100401 Manna et al., Phys. Rev. X 8 (2018) 041045, arXiv:1712.10174 Manna et al., Nature Review Materials, 3 (2018) 244 arXiv:1802.02838v1 high through put

playing with symmetry:

Noky et al., in submitted Heusler and Berry

푀 2 휌푥푦 = a푥푥 + b푥푥 푴. ??? outline

• Introduction • Topological insulators • Weyl semimetals – broken symmetry • Magnetic Weyl semimetals • New Fermions • Outlook – axion insultors and beyond new Fermions

6-fold crossings

8-fold degenerate points new Fermions

Ag3Se2Au, Ba4Bi3

• Chiral Crystals B20, Skyrmions, CoSi, MnSi, PdGa, RhSi • Superconductors A15 superconductors: Nb3Sn, Li2Pd3B

Bradlyn et al., Science, 353, 6299, (2016) universe – on a lattice

Carlo Beenakker Commentary Heisenberg (1930): We observe space as a continuum, but we might entertain the thought that there is an underlying lattice and that space is actually a crystal. Which particles would inhabit such a lattice world? Werner Heisenberg Gitterwelt (lattice world) hosted electrons that could morph into protons, photons that were not massless, and more peculiarities that compelled him to abandon “this completely crazy idea” chiral topology molecules with different chiralities B20 type cubic noncentrosymmetric crystallographic have different properties structure described by the chiral P213 space group nr 198 chiral topology

chiral crystals optical activity magnetochiral anisotropy … topological crystals unusual surface states large photogalvanic effect … new Fermions largest Fermi arc unusual phonon dynamics Gyrotropic magnetic effect Quantized circular photogalvaniceffect … chiral topology

Polycrystal PtAl, Homochiral PdGa Single crystals CoSi, MnSi, FeGe, RhSi, PtGa

RhSi

PtGa 6 fold 4 fold chiral topology

PtAl

Schröter et al. Nature Physics online 2019, preprint arXiv: 1812.03310 Fermi arcs

Chiral copies of the Fermi arcs

Schröter et al. Nature Physics online 2019, preprint arXiv: 1812.03310 Fermi arcs

Sanchez et al. Nature online 2019, preprint arXiv:1812.04466 Fermi arcs

Chiral copies of the Fermi arcs

Schröter et al. preprint arXiv: 1907.08723 quantized circular photogalvanic effect

CPGE trace is effectively quantized in terms of the fundamental constants e, h, c and e0 with no material-dependentparameters. This is so because the CPGE directly measures the topological charge of Weyl points, and non-quantized corrections from disorder and additional bands can be small over a significant range of incident frequencies. quantized circular photogalvanic effect

Excitation of Weyl fermions - a current that is quantized in units of material-independent fundamental constants over a range of photon energies

Frequency-independent plateau at low photon energy abruptly falls-off above 0.66 eV

Reez et al. under review, preprint arXiv:1812.04466 What comes next? correlated topological materials

SrPd3O4 Ta2Se8I

12 5x10-5

10 Ta Se I 2 8 -5 ) 4x10

) Ta2Se8I m

8 m



 ( 0 T ( 3x10-5

6 9 T

CDW 260K -5

4 2x10

Resistivity Resistivity 2 1x10-5

0 0 50 100 150 200 250 300 220 240 260 280 300 Temperature (K) Temperature (K) with Binghai Yan, Jansen with Bernevig, Sun, Gooth axion insulator

• The axion is a hypothetical elementary particle postulated by the Peccei–Quinn theory in 1977 to resolve the strong CP (combined symmetries of charge conjugation and parity) problem in quantum chromodynamics (QCD). If axions exist and have low mass within a specific range, they are of interest as a possible component of cold dark matter. Wikipedia correlated topological materials

Ta 2Se8I

12 5x10-5

10 Ta Se I 2 8 -5 ) 4x10

) Ta2Se8I m

8 m



 ( 0 T ( 3x10-5

6 9 T

CDW 260K -5

4 2x10

Resistivity Resistivity 2 1x10-5

0 0 50 100 150 200 250 300 220 240 260 280 300 Temperature (K) Temperature (K) Gooth et al. Nature accepted, preprint arXiv:1906.04510 axion insulator

Gooth et al. Nature accepted, preprint arXiv:1906.04510 axion insulator

add a magnetic moment more axion … catalysis

Hydrogen evolution

800 NbP

NbP 300 TaP )

NbP-m ) TaP-m 600

200

moles/g



 ( 400 ( Hydrogen evolution in 100 magnetic field (m=3000

200

evolved

evolved 2 2 Gauss)

H H 0 0 1 2 3 0 1 2 3 Time(hours) Time(hours)

C. R. Rajamathi, et la., Advanced Materials 29 (2017) 1606202 Meiser Nature Reviews Materials 2017, doi:10.1038/natrevmats.2017.21 semimetals and 3D QHE vision new chemistry and physics - New topological materials beyond the single particle picture - high mobility, topological surface states for catalysis - new emergent properties beyond condensed matter physics - inspired by astrophysics and high energy physics potential applications - spintronics H - quantum computing H2O 2 - energy conversion - catalysis … Surface states

Chiral electrons thank you for your attention

Andrei Yulin Chen Bernevig Stuart Parkin and team and teams