Topological Dirac & Weyl fermions in optical properties

Transcription

1 Topological Dirac & Weyl fermions in optical properties Friedhelm Bechstedt 1 L. Matthes 1, S. Küfner 1, J. Furthmüller 1 A. Mosca Conte 2, D. Grassano 3, O. Pulci 2,3 1 Friedrich-Schiller-Universität Jena, Germany 2 Mediterranean Institute for Fundamental Physics, Rome, Italy 3 Università di Roma Tor Vergata, Rome, Italy

2 History / Introduction Dirac fermions Standard model: all fermions (including neutrinos with mass) P.A.M. Dirac, Proc. Roy. Soc. London Ser. A 117, 610 (1928); 118, 351 (1928) Weyl fermions Massless fermions in Dirac equation H. Weyl, Z. Phys. A 56, 330 (1929) Discovery as low-energy excitations in solids = Weyl semimetals, which band disperse linearly in 3D k-space through a touching/crossing point called Weyl node Realization in solids: Band overlap / inversion (generalized) Hamiltonian (from k p theory) ˆ ˆ ˆ H = ( vp 0 ) σ 0 + p( v F σ) tilting anisotropic Dirac cones S.-Y. Xu et al., Science 349, 613 (2015) B. Lv et al., Phys. Rev. X 5, (2015)

3 Trend in band overlap: N-(8-N) semiconductors (E g, a 0 ) mass-darwin (cation s) inversion so = 0.4 ev α-sn so = 0.8 ev Criterium: toward inverted gap so = 0.9 ev so = 0.8 ev spin-orbit (anion p)

4 Band inversion: Spin-orbit coupling (SOC) band inversion (scalar relativistic effects) SOC opens gap = topological insulator (TI) with metallic surface states (Dirac cones) B. Yan, C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017) SOC makes gap except at isolated linearly crossing Weyl point (pairwise instead Dirac point) Dirac semimetal (DSM) Weyl semimetal (WSM) with exotic surface states (Fermi arcs)

5 Benefits / Applications Discovery of elementary particles predicted 80 years ago Explanation of dark matter by Dirac, Majorana and Weyl fermions P.B. Pal, Am. J. Phys. 79, 485 (2011) Super (Veselago) lenses R.D.Y. Hills et al., PRB 95, (2017) Electronic devices Qubits Anyons (fast carriers) A.G. Grushin, J.H. Bardarson, Physics 10, 63 (2017) (magnetic monopoles) D. Castelvecchi, Nature 547, 272 (2017)

6 Outline here: focus on IR optics Illustration: Dirac physics in 2D Dirac physics in 3D: Dirac semimetals Weyl physics in 3D: Weyl semimetals

7 Modeling: Atomic geometries ^= Combination of repeated slab / supercell approach with DFT (LDA, PAW) Electronic structures = ^ with SOC approximate quasiparticle effects: e.g. HSE06 Atomic relaxation, real surfaces / interfaces, including bonding / selfconsistency but L large enough

8 (Ab initio) Modeling: Optical properties density functional theory geometry, electronic bands, and wave functions dielectric function (ω) (independent-particle approximation) optical propertiesvia optical conductivity 0 [ ] σω ( ) = εω i ( ω) 1 Compensation of QP blue shift and excitonic red shift

9 Dirac physics in 2D

10 2D topological insulators 2D insulator with conducting edge states metallic boundaries from topological behavior topological states protected by timereversal symmetry (TRS) protected : cannot be removed by edge passivation spin-polarized

11 Model system: X-ene 2D honeycomb material beyond graphene? Graphene Other 2D materials? Silicene, germanene, stanene (graphene-like Si, Ge, Sn) Theory: energetically stable with similar electronic properties as graphene K. Takeda, K. Shiraishi, PRB 50, (1994) G. G. Guzmán-Verri et al., PRB 76, (2007) but: other 2D arrangements F. Matusalem et al., PRB 92, (2015)

12 Group-IV 2D honeycomb crystals Top view Point-group symmetry D 3d (with SOC D 3 ) Side view = 0.44 Å Experiment: P. Vogt et al., PRL 108, (2012)

13 Quasiparticle band structures graphene silicene germanene stanene 2D Dirac cones at K despite sp 3 and buckling opening of small (large) gaps due to SOC (functionalization) L. Matthes,, O. Pulci, F.B., J. Phys. CM 25, (2013)

14 Band structure and optical matrix elements normalized to v F Interband energies and transition matrix elements near K (K ) independent of group-iv element if normalization with v F Allowed optical transition graphene silicene germanene F. Bechstedt,, O. Pulci, APL 100, (2012) L. Matthes, O. Pulci, F.B., PRB 87, (2013)

15 Absorbance: Independent-particle approximation germanene silicene graphene A(0) = πα A(ω) A(ω) independent of buckling, hybridization, element, and gauge quadratic increase with ω deviations near van Hove singularities F. Bechstedt,., O. Pulci, APL 100, (2012) L. Matthes, O. Pulci, F.B., PRB 78, (2013)

16 Infrared absorbance A(ω) of graphene R.R. Nair, 1 P. Blake, 1 A.N Grigorenko, 1 K.S. Novoselov, 1 T.J. Booth, 1 T. Stauber, 2 N.M.R. Peres, 2 A.K. Geim 1 * Science 320, 1308 (2008) AA = ππππ 2.3% (Sommerfeld fine structure constantα)

17 Band topology: Toplogical invariant (with inversion symmetry) Time-reversal-invariant moment (TRIM) k = k + G i i L. Fu, C.L. Kane, PRB 76, (2007) matrix element of parity operator P i ξ ( ) 2 k = nk P nk n i i i Germanene δ N ( k ) = i n= 1 ξ i 2n i ξ 2n Products of parities : N = 4 occupied bands Order parameter ν (= Z 2 ): 4 ν ( ) ( k ) ( ) ( M ) 1 = δ i =δ Γ δ = 1 i= 1 Ζ 2 = ν = 1 ^ quantum spin Hall phase = ^ toplogical insulator 3

18 Frequency-dependent spin Hall conductivity Kubo formula z (spin) J x s charge current spin current jx ( ) j spin ( x)~curl sx ( ) topology spin Berry phase Ω ν (k,ω) SH σxy ( ω ) = e h f( εν( k)) Ων( k, ω) V 2 1 k ν J y, E y F.D.M. Haldane, PRL 60, 635 (1988) C.L. Kane, E.J. Mele, PRL 95, (2005) electrons of different spin (J xs ) flow charge current (J y ) quantization (QSH effect) for µ, kt << so

19 Quantum spin Hall effect: functionalized germanene GeI, Ge, GeH GeI geometry band structure spin Hall conductivity Quantization e 2 /h: TI (GeI, Ge) vanishing effect: CI (GeH) L. Matthes,, F.B., PRB 93, (R) (2016)

20 2D summary: General: Topolog. insulator & quantum spin Hall phase ruled by relativistic effects (band inversion) Model (graphene-like honeycomb group-iv layer) with Dirac cones Absorbance ruled by finestructure constant massless Dirac fermions TI in dependence of functionalization QSH phase depending on gap massive Dirac fermions (gap)

21 3D Dirac/Weyl physics: Topological semimetals Dirac SM Weyl SM (TR, I) (TR, without I) Cd 3 As 2 TaAs, TaP, BiNa 3 NbAs, NbP

22 NEW MATERIALS: Weyl/Dirac 3D fermions Cd 3 As 2 Dirac semimetal TaAs, TaP, NbAs, NbP Weyl semimetals

23 H Topological semimetal Dirac fermions time-reversal & inversion symmetry =γ γ k ˆ non-chiral twofold Kramers degenerate symmetry protected unstable Weyl fermions symmetry break H = k σˆ chiral non-degenerate topologically protected stable against perturbation Perturbation no time-reversal no inversion

24 3D IR optical properties linear bands (Dirac or Weyl fermions) ω 0 Im ε = constant/v F, Re σ ~ αω/v F in contrast to α = e 2 /hc ~ 1/137 2D (graphene, silicene, germanene ) Re σ πα

25 3D Dirac fermions

26 Cd 3 As 2 geometry: top Body-centered tetragonal structure with inversion symmetry As Cd 6.46 A Small cube building block Anti-fluorite structure 2 Cd vacancies

27 Band structure (bct, 80-atom primitive cell) Dirac touching point A. Mosca Conte, O.P., F.B., Scientific Reports 7, (2017) two touching points (despite SOC, symmetry-protected) Dirac cones? inversion symmetry, therefore Kramers degeneracy, like graphene pseudolinearity away Fermi level

28 near Fermi level DFT electronic bands with SOI more zero-gap semiconductor strongly anisotropic Dirac cones along tetragonal axis

29 Orbital character Only As p orbitals (at odd with small cube findings) 3D graphene analogue

30 Dirac fermions? ARPES: Z.K. Liu et al. Nat. Mat. 13, 677 (2014), M. Neupane et al., Nat. Commun. 5, doi: /2014 S. Borisenko et al., PRL 113, (2014) low-energy Dirac electrons high-energy Kane electrons

31 Optical conductivity Im ε tends to a plateau optical conductivity linear near 3D Dirac cone (in 2D constant) 1 ωc Re σjj( ω ) = α 12π v Fj Dirac & Kane electrons (>0.1 ev due to pseudo-linear bands) NOT to Dirac cone A. MoscaConte, O. Pulci, F. Bechstedt, Sci. Rep. 7, (2017)

32 Exp. optical conductivity Good agreement with theory but wrong interpretation? D. Neubauer et al., Phys. Rev. B 93, (R) (2016)

33 Summary: Dirac semimetal bct-cd 3 As 2 2 Dirac points near Γ at ΓΖ and -ΓZ asymmetric Dirac cones along tetragonal axis only low-energy excitations (< 0.1 ev) : Dirac higher energies quasi-linearity (> 0.1 ev) ^= Kane electrons? Fermi velocities ~ graphene dangling As p point toward Cd vacancies π-electron system linear optical conductivity ~ α

34 Weyl fermions in 3D

35 Transition metal monopnictides: structure crystallize in body-centered tetragonal (bct) structure non-symmorphic space group I4 1 md NO INVERSION SYMMETRY TaAs TaP NbAs NbP *H. Boller et al. - Acta Crystallographica (1963) *Schönberg, N. I. L. S. - Acta Chem. Scand 8.2 (1954)

36 Transition metal monopnictides (reciprocal space) without spin-orbit: line nodes on mirror planes with spin-orbit: 24 Weyl nodes (4 W1 and 8 W2 pairs)

37 Band structure trivial no SOC-induced splittings at Weyl nodes cation d states Weyl and Kane electrons?? D. Grassano, O. Pulci, F.B., Sci. Reports 8, 3534 (2018)

38 Position of Weyl nodes Measured W1 position: Å -1 Computed W1 position: Å -1 k W1 = (0.925, ± 0.014, 0) Å -1 Souma, S., Wang, Z., Kotaka, H., Sato, T., Nakayama, K., Tanaka, Y., Kimizuka, H., Takahashi, T., Yamauchi, K., Oguchi, T. and Segawa, K., Phys. Rev. B 93, (2016)

39 Pairs of tilted anisotropic Weyl nodes (TaAs) Fermi level (4 pairs) pairing anisotropic tilted no gap (8 pairs) D. Grassano, O. Pulci, A. Mosca Conte, F. Bechstedt, Sci. Reports 8, 3534 (2018)

40 ARPES of Weyl fermions (TaAs) W2 (electron pocket) W1 (hole pocket) Su-Yang Xu et al., Science 349, 613 (2015)

41 Tilt & anisotropy 10 5 m/s v xx v yy v zz v xy v xz v yz v 0x v 0y v 0z W1 TaAs W2 TaAs Hamiltonian = ( vp ˆ ˆ 0 ) σ 0 + ( vp j ) σ anisotropic deformation of Dirac cones tilt vector j= xyz,, chirality = sgn (det (v - ij)) j no tilt, no anisotropy strong tilt, e.g NbP but no type-ii B. Yan, C. Felser, Annu. Rev. Condens. Matter Phys. 8, 337 (2017)

42 Measured optical conductivity Attention: via Kramers-Kronig (assuming constanthighenergy reflectivity) B. Xu et al., Phys. Rev. B 93, (2016)

43 Optical conductivity (theory) Besides Drude term: roughly linear frequency increase up to saturation Details: deviations due to - electron / hole occupation - kind of Weyl point - mixture, not only spectral density Weyl fermionshardlyvisible onset & contribution depend on W1/2 D. Grassano, O. Pulci, F. Bechstedt, Sci. Rep. 8, 3534 (2018)

44 Outlook: Fermi arcs γ k space Zak phase Chern number γ ( k ) ( ) ( k ) C k Ò = dfω( k) ( ) F k 1 k < =γ /2π= 0 k > k k 0 0 topological surface states along connection line with dispersion this line & surface normal How they modify surface optical properties?

45 Summary: Weyl semimetals Novel 3D topological semimetals Dirac (time-rev. & inversion) Weyl (time-rev.): bct-taas (TaP, NbAs, NbP) 12 pairs of W1 and W2 Weyl nodes expected linear optical conductivity modified - Drude - occupation cut - mixture of occupation and JDOS - non-topological points difficult to identify Weyl fermions Fermi arcs in optics?