Weyl and Transport. Claudia Felser. Nature
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1 Weyl and Transport Claudia Felser Nature
2 Graphene Graphene s conduc2vity exhibits values close to the conduc2vity quantum e2/h per carrier type Graphene s charge carriers can be tuned con2nuously between electrons and holes in concentra2ons n = cm 2 Mobili2es μ can exceed 15,000 cm 2 V 1 s 1 under ambient condi2ons InSb has μ 77,000 cm 2 V 1 s 1 Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183 (2007).
3 Graphene Geim, A. K. & Novoselov, K. S. The rise of graphene. Nature Mater. 6, 183 (2007).
4 Particles Universe Condensed matter Quantum field theory Berry curvature Dirac Cd 3 As 2 Guido Kreiner Higgs YMnO 3 Lichtenberg Spaldin Weyl TaAs Vicky Süss, Marcus Schmidt Majorana YPtBi Chandra Shekhar
5 Family of Quantum Hall Effects 1985 Klaus von Klitzing 1998 Horst Ludwig Störmer and Daniel Tsui 2010 Andre Geim and Konstan2n Novoselov S Oh Science 340 (2013) David Thouless, Duncan Haldane und Michael Kosterlitz
6 Hall effect Edwin Herbert Hall Lorentz force deflect charge par2cles We measure the resistance without and with a magne2c field Metal, semiconductor, or insulator Electron or hole conduc2vity Resistance in a magne2c field: Magnetoresistance A. Husmann et al., Nature 417 (2002) 421
7 Hall effect ρ yx (arb. unit) Si µ 0 H (arb. unit)
8 Anomalous-Hall effect Ferromagne2c materials have internal magne2c field No effect so far in an2ferromagnets AHE scales with the Magne2za2on External effects Much larger than normal Hall Effect, but not well understood ; Possible Berry-phase effect ρh = R0 B + 4πR s M Berry curvature: M.V. Berry, Proc. Royal Soc. London (1984)
9 Spin-Hall effect Separa2on of electron spins without magne2c field Spin-current genera2on in nonmagne2c systems Predicted in 1971 or 2004? Observed in semiconductors (2004) and metals Extrinsic or intrinsic? Experiment (2004): Kato, Myars, Gossard, Awschalom, Science 306, 1910 Theory D'yakonov & Perel', Exp. Theor. Phys. Lem. (1971) INTRINSIC SPIN-HALL EFFECT: Murakami et al Science 2003 (cond-mat/ ) Sinova et al PRL 2004 (cont-mat/ ) Berry curvature: M.V. Berry, Proc. Royal Soc. London (1984)
10 Anomalous Hall effect Intrinisic Hall Berry curvature Magne2sa2on? Extrinsic Hall Skew scamering Side jumps
11 The Spin Hall effect (SHE) à Spin Orbit Torque (SOT) Posi2ve Spin Hall Angle (SHA) Spin Hall effect à Spin orbit torque Spin current flows across the interface and applies a torque on the FM layer Charge to spin conversion efficiency Miron et al., Nature (2011) Liu et al., Science (2012) Ta, Pt, IrMn, Bi 2 Se 3 Weyl semi-metal
12 Fundamental limits: switching current à How many spin polarized electrons needs to switch nano-element? à Nano-element: 10 x 10 x 1 nm 3 - need ~100,000 electrons to switch magnetization - requires that current (µa) x time interval (nsec) ~ 30 µa.nsec à 10 µa for 3 nsec à To reduce current could use orbital moments e.g. using spin-orbit coupling à Lower limit: - need to overcome energy barrier that stabilizes MTJ against thermal fluctuations - some angular momentum lost to lattice (damping)
13 Dirac and Weyl semimetals Paul Klee
14 Bohm-Jung Yang and Naoto Nagaosa, arxiv: Dirac semimetals
15 3D Dirac Cd 3 As 2 Arthur J. Rosenberg and Theodore C. Harman Journal of Applied Physics 30, 1621 (1959) Wang, Z. J., Weng, H. M., Wu, Q. S., Dai, X. & Fang, Z., Phys. Rev. B 88, (2013). Liu, Z. K. et al. Nature Mater. 13, (2014).
16 Electronic structure CdTe HgTe ScPtSb ScPtBi Chadov, Qi, Kübler, Zhang, Felser Nature Mat. 9 (2010) 541, arxiv:
17 Application Spin Hall Effect K. Chadova, et al., Phys. Rev. B 93 (2016) preprint: arxiv:
18 Application Spin Hall Effect K. Chadova, et al., Phys. Rev. B 93 (2016) preprint: arxiv:
19 The layered Heusler
20 Weyl Semimetals Breaking symmtery NbP
21 Weyl Semimetal Breaking symmetry E with Inversion symmetry (Structural distor2on) Breaking 2me reversal symmetry Magne2c field E without k Dirac points are at high symmetry points Weyl points are not at high symmetry points k
22 Weyl semimetals a TI SOC Weyl points Band inversion WSM DSM b Dirac point C = 0 C = 1
23 C = 0 C = 1 c d Hole Electron Type-I WSM Type-II WSM Graphene Alexey A. Soluyanov, et al., Nature 527, 495 (2005) A. K. Geim, A. H. MacDonald Physics Today, 08.(2007), Shekhar, et al., Nature Physics 11 (2015) 645
24 Weyl semimetals 3D topological Weyl semimetals - breaking 2me reversal symmetry in transport measurement we should see: 1. Fermi arc 2. Chiral anomaly 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) AA Burkov, L Balents, PRL (2012)
25 2 Viola2on of chiral symmetry. Quantum anomaly in QED In Quantum Electro Dynamics (rela2vis2c quantum field theory) chiral charge conserva2on can be violated for massless fermions! Pion decay: " Neutral Pion $ Photon Photon Electric charge: 0 0 Spin: half-integer integer Chiral anomaly Adler, S. L. Phys. Rev. 177, 5 (1969). Bell, J. S. & Jackiw, R. Nuovo Cim. A60, 4 (1969)
26 Weyl semimetals in non-centro NbP NbP, NbAs, TaP, TaAs Shekhar, et al., Nature Physics 11 (2015) 645, Frank Arnold, et al. Nature Communica2on 7 (2016) Weng, et al. Phys. Rev. X 5, (2015) Huang et al. preprint arxiv:
27 NbP, TaP, TaAs Increasing spin orbit coupling increases heavier elements Distance between the Weyl points increases NbAs TaP Z. K. Liu et al., Nature Mat. 15 (2016) 27 Yang, et al. Nature Phys. 11 (2015) 728
28 Resistance Measurement We measure the resistance without and with a magne2c field Metal, semiconductor, or insulator Electron or hole conduc2vity Resistance in a magne2c field: Magnetoresistance
29 Weyl Points in non-centro NbP NbP is a topological Weyl semimetal with massless rela2vis2c electrons extremely large magnetoresistance of 850,000% at 1.85 K, 9T (250% at room temperature) an ultrahigh carrier mobility of 5*10 6 cm 2 / V s Shekhar, et al., Nature Physics 11 (2015) 645, Frank Arnold, et al. Nature Communica2on 7 (2016) Weng, et al. Phys. Rev. X 5, (2015) Huang. et al. preprint arxiv:
30 TaP Quantum Oscillations SQUID-VSM magne2za2on In c-direc2on the diamagne2c magne2za2on is superimposed by quantum oscilla2ons star2ng at 0.6 T In a-direc2on only weak quantum oscilla2ons are visible The frequency propor2onal to the extremal Fermi surface perpendicular to B
31 NbP and the Fermi surface Klotz et al. Phys. Rev. Lem. 2016
32 Chiral Anomaly Ga-doping relocate the Fermi energy in NbP close to the W2 Weyl points Therefore we observe a nega2ve MR as a signature of the chiral anomaly the, NMR survives up to room temperature Anna Corinna Niemann, Johannes Gooth et al. Scien2fic Reports 7 (2017) doi: /srep4339 preprint arxiv:
33 Chiral Anomaly Ga-doping relocate the Fermi energy in NbP close to the W2 Weyl points Anna Corinna Niemann, Johannes Gooth et al. Scien2fic Reports 7 (2017) doi: /srep4339 preprint arxiv:
34 Longitudinal magneto-transport E B 3 The PMC is locked to E B, as expected for Chiral anomaly. [Shekhar, C. et al. Nat. Phys. 11, 3372 (2015)]
35 Chiral Anomaly Experimental signatures for the mixed axial-gravita2onal anomaly in Weyl semimetals In solid state physics, mixed axial-gravita2onal anomaly can be iden2fied by a posi2ve magneto-thermoelectric conductance (PMTG) for ΔT ll B. Low fields: quadra2c High fields: deminishes B ΔT ll B dictates sensi2vity on alignement of B and ΔT. G T 3 [Lucas, et al. Proceedings of the Na;onal Academy of Sciences 113, 9463 (2016)]
36 Gravitational Anomaly Landsteiner, et al. Gravita2onal anomaly and transport phenomena. Phys. Rev. Lem. 107, (2011). URL Jensen, et al. Thermodynamics, gravita2onal anomalies and cones. Journal of High Energy Physics 2013, 88 (2013). Lucas, A., Davison, R. A. & Sachdev, S. Hydrodynamic theory of thermoelectric transport and nega2ve magnetoresistance in weyl semimetals. PNAS 113, (2016). A posi2ve longitudinal magneto-thermoelectric conductance (PMTC) in the Weyl semimetal NbP for collinear temperature gradients and magne2c fields that vanishes in the ultra quantum limit. Johannes Gooth et al. Experimental signatures of the gravita2onal anomaly in the Weyl semimetal NbP, Nature accepted arxiv:
37 Gravitational Anomaly Landsteiner, et al. Gravita2onal anomaly and transport phenomena. Phys. Rev. Lem. 107, (2011). URL Jensen, et al. Thermodynamics, gravita2onal anomalies and cones. Journal of High Energy Physics 2013, 88 (2013). Lucas, A., Davison, R. A. & Sachdev, S. Hydrodynamic theory of thermoelectric transport and nega2ve magnetoresistance in weyl semimetals. PNAS 113, (2016). A posi2ve longitudinal magneto-thermoelectric conductance (PMTC) in the Weyl semimetal NbP for collinear temperature gradients and magne2c fields that vanishes in the ultra quantum limit. Johannes Gooth et al. Experimental signatures of the gravita2onal anomaly in the Weyl semimetal NbP, Nature accepted arxiv:
38 Hydrodynamics Hydrodynamic electron fluid is defined by momentum-conserving electronelectron scamering Viola2on of Wiedeman-Franz law Viscosity-induced shear forces making the electrical resis2vity a func2on of the channel width
39 Hydrodynamics Experimental evidence that the resistance of restricted channels of the ultra-pure two-dimensional metal PdCoO2 has a large viscous contribution
40 High mobility wires
41 A Bemer Weyl Semimetals
42 Predic2on: G. Autès, et al.; Phys. Rev. Lem. 117 (2016) WP 2 protected Weyl
43 WP 2 protected Weyl Predic2on: G. Autès, et al.; Phys. Rev. Lem. 117 (2016) Extremely high magnetoresistance and conduc2vity in the type-ii Weyl semimetal WP2, Nitesh, et al.; arxiv:
44 WP 2 protected Weyl Predic2on: G. Autès, et al.; Phys. Rev. Lem. 117 (2016) Extremely high magnetoresistance and conduc2vity in the type-ii Weyl semimetal WP2, Nitesh, et al.; arxiv:
45 ARPES and the Band Structure Photoemission, Nan Xu, Ming Shi, Paul Scherrer Ins2tute, Swiss Light Source, CH-5232 Villigen PSI, Switzerland. Extremely high magnetoresistance and conduc2vity in the type-ii Weyl semimetal WP2, Nitesh, et al.; arxiv:
46 WP 2 protected Weyl Magnetotransport in a novel Weyl WP 2 a 10-5 ρ (Ω cm) T 0.1 T 0.3 T 0.5 T 1 T 2 T 3 T 5 T 7 T 9 T c T (K) d MR (%) 4x10 6 3x10 6 2x10 6 1x RRR = RRR = 8275 RRR = x x µ 0 H (T) 2 K 1.2 x10 6 MR (%) MR B 1.94 fit µ 0 H (T) Predic2on: G. Autès, et al.; Phys. Rev. Lem. 117 (2016) Extremely high magnetoresistance and conduc2vity in the type-ii Weyl semimetal WP2, Nitesh, et al.; arxiv:
47 Macroscopic Mean Free Path Compound ρ (Ωcm) l (µm) µ (cm 2 V -1 s -1 ) n (cm -3 ) MoP WP WC ~ PtCoO PdCoO x WC J. B. He et al. arxiv: Pallavi Kushwaha, et al. Sci. Adv.1 (2015) e P. Moll Science 351, (2016) 1061 Chandra Shekhar et al. arxiv: Nitesh, et al.; arxiv:
48 Hydrodynamics In the ballistic regime (w << l er, l mr ): ρ ~ w -1 Hydrodynamic effects become dominant electron-electron scamering l er << w << l mr, with electron-electron scamering length l er = v F % er w the sample width, l mr = v F % mr the mean free path and v F the Fermi velocity In the Navier-Stokes flow limit: ρ = m * /(e 2 n) 12ηw -2 R. N. Gurzhy, A. N. Kalinenko, A. I. Kopeliovich, Hydrodynamic effects in the electrical conduc2vity of impure metals. Sov. Physics-JETP. 69, (1989). P. S. Alekseev, Nega2ve magnetoresistance in viscous flow of two-dimensional electrons. Phys. Rev. LeD. 117 (2016). T. Scaffidi, N. Nandi, B. Schmidt, A. P. Mackenzie, J. E. Moore, Hydrodynamic Electron Flow and Hall Viscosity. Phys. Rev. LeD. 118, (2017).
49 Water, Gas or Electrons
50 Hydrodynamic flow J. Gooth et al. submimed, arxiv:
51 a Hydrodynamic flow 10-5 ρ (Ω cm) T 0.1 T 0.3 T 0.5 T 1 T 2 T 3 T 5 T 7 T 9 T c d MR (%) MR B 1.94 fit µ 0 H (T) P. J. W. Moll et al., Science /science.aac8385 (2016). J. Gooth et al. submimed, arxiv:
52 Hydrodynamic flow a 10-5 ρ (Ω cm) c T 0.1 T 0.3 T 0.5 T 1 T 2 T 3 T 5 T 7 T 9 T Hydrodynamic electron fluid <15K d conven2onal metallic state at T higher 150K MR (%) MR B 1.94 fit The hydrodynamic regime: a viscosity-induced dependence of the electrical resis2vity on the square of the channel width µ 0 H (T) ρ = m * /(e 2 n) 12ηw -2 a strong viola2on of the Wiedemann-Franz law J. Gooth et al. submimed, arxiv:
53 a Hydrodynamic flow ρ (Ω cm) T 0.1 T 0.3 T 0.5 T 1 T 2 T 3 T 5 T 7 T 9 T PdCoO 2 c d MR (%) MR B 1.94 fit µ 0 H (T) P. J. W. Moll et al., Science /science.aac8385 (2016) R. Daou, et al. and Antoine Maignan, Phys. Rev. B 91, (R) (2015) J. Gooth et al. submimed, arxiv:
54 Giant Nernst Topology - Hydrodynamic Ramzy Daou, Raymond Frésard, Sylvie Hébert, and Antoine Maignan, Phys. Rev. B 92, (2015) Sarah J. Watzman, et al. preprint arxiv:
55 Magnetohydrodynamics, Planckian bound of dissipation Grey dots: the magnetohydrodynamic model in the Navier-Stokes flow limit Momentum relaxa2on 2mes t mr thermal energy relaxa2on 2mes t er, Dashed line marks the Planckian bound on the dissipa2on 2me % ħ = ħ/ (' $ (). J. Gooth et al. submimed, arxiv:
56 Viscosity of the electron fluid in WP 2 The dynamic viscosity is η D = kgm -1 s -1 at 4 K.
57 TaAs 1.0 Reflectivity 0.9 TaAs T=7K 0 T 7 T σ 1 (Ω -1 cm -1 ) These fringes are measurement artefacts 2000 Dashed lines are guides to the eye Data below 50 cm -1 are less reliable Huang et al. Phys. Rev. X 5, (2015) Freuency (cm -1 ) A. Pronim unpublished
58 Design scheme: Dirac and Weyl Semimetal Band inversion e.g. inert pair effect Crossing band due to enforced degenera2on New quantum effects electron liquid Hoffmann Angewandte. Chem. 26 (1987) 846
59 Weyl Semimetals Magne2cally induced
60 Predicting topological insulators ScNiSb LaPtBi LaPtBi S. Chadov et al., Nat. Mater (2010). H. Lin et al., Nat. Mater (2010).
61 REPtBi multifunctional topologic insulators Mul2func2onal proper2es RE: Gd, Tb, Sm Magne2sm and TI RE: Ce An2ferromagne2sm with GdPtBi complex behaviour of the Fermi surface RE: Yb Kondo insulator and TI YbPtBi is a super heavy fermion with the highest γ value Pt RE Bi (+f n ) + 5 = 18 YPdBi YPtBi S. Chadov et al., Nat. Mater. 9, 541 (2010). H. Lin et al., Nat. Mater. 9, 546 (2010). GdPtBi LuPtBi
62 Weyl semimetals 3D topological Weyl semimetals - breaking dme reversal symmetry in transport measurement we should see 1. Intrinsic anomalous Hall effect 2. Chiral anomaly 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) AA Burkov, L Balents, PRL (2012)
63 Weyl GdPtBi in a magnetic field a Y, Nd, Gd Pt Bi b [111] c d e [111] 1 E F Γ 8 Γ 6 Exchange field L chirality Energy (ev) chirality YPtBi f electrons GdPtBi 1 W Γ L X C. Shekhar et al., arxiv: , (2016). M. Hirschberger et al.,. Nature Mat. Online arxiv: , (2016).
64 Chiral Anomaly neg. quadratic MR 2K Claudia Felser and Binghai Yan, Nature Materials 15 (2016) 1149 C. Shekhar et al., arxiv: , (2016). M. Hirschberger et al. Nature Mat. online, arxiv: , (2016).
65 GdPtBi Anomalous Hall Effect In Ferromagnets an AHE scales with the magne2c moment An2ferromagnets show no AHE A Hall angle of 0.2 is excep2onal high ρh = R0 B + 4πR s M Shekhar et al., arxiv: , (2016). T. Suzuki, & J. G. Checkelsky, Nature Physics (2016) doi: /nphys3831
66 Chiral Anomaly neg. quadratic MR Claudia Felser and Binghai Yan, Nature Materials 15 (2016) 1149 C. Shekhar et al., arxiv: , (2016). M. Hirschberger et al. Nature Mat. online, arxiv: , (2016).
67 GdPtBi Anomalous Hall Effect ρ yx (arb. unit) GdPtBi MnSi µ 0 H (arb. unit)
68 Do Antiferromagnets have AHE Berry Phase ρh = R0 B + 4πR s M No! No Berry phase
69 Heusler Weyl and Berry
70 Materials: to half metallic ferromagnets X 2 YZ Example: Co 2 MnSi magic valence electron number: 24 valence electrons = 24 + magnetic moments Co 2 MnSi: = 29 Ms = 5µ B Kübler et al., PRB 28, 1745 (1983) de Groot RA, et al. PRL (1983) Galanakis et al., PRB 66, (2002)
71 Weyl semimetals in Heusler compounds Zhijun Wang, et al., arxiv: Guoqing Chang et al., arxiv: Barth et al. PRB 81,
72 AHE in half metallic ferromagnets The AHE depends only on the Berry curvature in Heusler compounds and not on the magnedsadon Kübler, Felser, PRB 85 (2012) Vidal et al Appl.Phys.Lem. 99 (2011) Kübler, Felser, EPL 114 (2016) ρh = R0 B + 4πR s M
73 AHE in half metallic ferromagnets with Weyl Giant AHE in Co 2 MnAl σ σ xy xy = 1800 S/cm 2000 S/cm calc. meas. Kübler, Felser, PRB 85 (2012) Vidal et al Appl.Phys.Lem. 99 (2011) Kübler, Felser, EPL 114 (2016) Weyl points are the origin for a large Berry phase and a Giant AHE
74 Heusler, Weyl and Berry Can we design an AFM with a Berry curvature
75 Hexagonal Antiferromagnet Chen, Niu, and MacDonald, Phys. Rev. Lem., 112 (2014) Kübler and Felser EPL 108 (2014) 67001
76 Non-collinear AFM in metallic Mn 3 Ge The anomalous Hall conductivities are normally assumed to be proportional to magnetization Kübler and Felser EPL 108 (2014) Nayak et al. preprint: arxiv: , Science Advances 2 (2016) e
77 Non-collinear AFM Mn 3 Ge/Mn 3 Sn Nayak et al. preprint: arxiv: , Science Advances 2 (2016) e Kiyohara, Nakatsuji, preprint: arxiv: , Nakatsuji, Kiyohara, & Higo, Nature, doi: /nature15723
78 Neel Temperature And more Giant Nernst, has already shown switching Nayak et al. preprint: arxiv: , Science Advances 2 (2016) e Kiyohara, Nakatsuji, preprint: arxiv: ,
79 Hao Yang, et al., preprint: arxiv: Fermiarcs in the Weyl AFM
80 Application Spin Hall Effect Yan Sun et al., Physical Review Lemer 117 (2016)
81 Giant AHE in AFM R s M s Slope R 0 Co 2 MnAl ρ yx (arb. unit) Mn 3 Ge µ 0 H (arb. unit)
82 Heusler, Weyl and Berry Can we design an a Ferro/Ferrimagnet with a zero Berry curvature
83 E E E E f E f E f Half-metal Spin-gapless Semiconductor Magne2c Semiconductor
![Weyl or Spingapless kx = 0 kz = 0 Energy E - ε F [ev] 1 0-1 σ xy (10 2 Ω -1 cm -1 ) -20 2-10 0 10 20 up down Hall kx = 0 kz = 0-2 Energy E - ε F [ev] -10-5 0 5](/docs-images/83/88181502/images/84-3.jpg "10 Density of states n(e) [ev -1 ] -20-10 0 10 20 2 1 0-1 σ xy (10 2 Ω -1 cm -1 ) up down Hall 0.")
84 Weyl or Spingapless kx = 0 kz = 0 Energy E - ε F [ev] σ xy (10 2 Ω -1 cm -1 ) up down Hall kx = 0 kz = 0-2 Energy E - ε F [ev] Density of states n(e) [ev -1 ] σ xy (10 2 Ω -1 cm -1 ) up down Hall 0.12 ev 74 Ω -1 cm Density of states n(e) [ev -1 ]
85 Weyl or Spingapless σ xy (ohm -1 cm -1 ) H (T) H? [001] 2 K 50 K 100 K 150 K 200 K σ xy (ohm -1 cm -1 ) H [100] 2 K 50 K 100 K 200 K 300 K H (T)
86 Weyl or Spingapless 3000 H [100] σ xy (ohm -1 cm -1 ) K 50 K 100 K 200 K 300 K H (T)
87 More semiconductors 26 Mn2CoAl CoMn2Al CoFeCrAl CoMnCrSi CoFeVSi FeMnCrSb 18 V 3 Al 21 FeVTiSi CoVScSi FeCrScSi FeVTiSi FeMnScAl 28 CoFeMnSi
88 Anomalous Hall effect R s M s Slope R 0 Mn 2 CoGa Co 2 MnAl ρ yx (arb. unit) Mn 3 Ge µ 0 H (arb. unit)
89 Anomalous Hall effect
90 Single Crystals available MoSe2-xTex MoTe2-xSex MoTe2 (T /2H) YPtBi NdPtBi GdPtBi YbPtBi ScPdBi YPdBi ErPdBi GdAuPb TmAuPb AuSmPb AuPrPb AuNdPb AuScSn AuLuSn AuYSn ErAuSn EuAuBi CaAgAs KMgSb KMgBi KHgSb KHgBi LiZnAs LiZnSb AlPt GdAs CoSi MoP WP TaP NbP NbAs TaAs NbP-Mo NbP-Cr TaP-Mo TaAsP CrNb3S6 V3S4 Cd3As2 MnP MnAs YbMnBi2 Ni2Mn1.4In0.6 YFe4Ge2 Mn1.4PtSn CuMnSb CuMnAs Co2Ti0.5V0.5Sn Co2VAl0.5Si0.5 Co2Ti0.5V0.5Si Mn2CoGa Co2MnGa Co2Al9 Co2MnAl Co2VGa0.5Si0.5 Co2TiSn Co2VGa Co2V0.8Mn0.2Ga CoFeMnSi Bi2Te2Se Bi2Te3 Bi2Se3 BiSbTe2S BiTeI BiTeBr BiTeCl LaBi, LaSb GdBi, GdSb HfSiS Bi4I4 BaSn2 BaCr2As2 BaCrFeAs2 CaPd3O4 SrPd3O4 BaBiO3 PtTe2 PtSe2 PdTe2 PdSe2 OsTe2 RhTe2 TaTe2 NbTe2 WSe2 HfTe5 MoTe2 TaS2 PdSb2 CuxWTe2 FexWTe2 WTe2 Co0,4TaS2 Fe0,4TaS2 Ag2Se IrO2 OsO2 ReO2 WP2 MoP2 VAl3 Mn3Ge Mn3Ir Mn3Rh Mn3Pt DyIn3
91 Real space topology - Skyrmions (A) Bloch skyrmion (B) antiskyrmion (C) Neel skyrmion
92 Mn 1.4 Pt 0.9 Pt 0.1 Sn: Anti-Skyrmions T=300 K, H= 0.23 T µ 0 H (T) T (K) Nayak, et al. preprint arxiv: Bogdanov et al PRB 66, (2002)
93 Single Crystals available MoSe2-xTex MoTe2-xSex MoTe2 (T /2H) YPtBi NdPtBi GdPtBi YbPtBi ScPdBi YPdBi ErPdBi GdAuPb TmAuPb AuSmPb AuPrPb AuNdPb AuScSn AuLuSn AuYSn ErAuSn EuAuBi CaAgAs KMgSb KMgBi KHgSb KHgBi LiZnAs LiZnSb AlPt GdAs CoSi MoP WP TaP NbP NbAs TaAs NbP-Mo NbP-Cr TaP-Mo TaAsP CrNb3S6 V3S4 Cd3As2 MnP MnAs YbMnBi2 Ni2Mn1.4In0.6 YFe4Ge2 Mn1.4PtSn CuMnSb CuMnAs Co2Ti0.5V0.5Sn Co2VAl0.5Si0.5 Co2Ti0.5V0.5Si Mn2CoGa Co2MnGa Co2Al9 Co2MnAl Co2VGa0.5Si0.5 Co2TiSn Co2VGa Co2V0.8Mn0.2Ga CoFeMnSi Bi2Te2Se Bi2Te3 Bi2Se3 BiSbTe2S BiTeI BiTeBr BiTeCl LaBi, LaSb GdBi, GdSb HfSiS Bi4I4 BaSn2 BaCr2As2 BaCrFeAs2 CaPd3O4 SrPd3O4 BaBiO3 PtTe2 PtSe2 PdTe2 PdSe2 OsTe2 RhTe2 TaTe2 NbTe2 WSe2 HfTe5 MoTe2 TaS2 PdSb2 CuxWTe2 FexWTe2 WTe2 Co0,4TaS2 Fe0,4TaS2 Ag2Se IrO2 OsO2 ReO2 WP2 MoP2 VAl3 Mn3Ge Mn3Ir Mn3Rh Mn3Pt DyIn3
Heusler compounds: Tunable materials with non trivial topologies. Claudia Felser
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