Optics and magneto-optics electrons in conical bands
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1 Optics and magneto-optics electrons in conical bands Milan Orlita Laboratoire National des Champs Magnétiques Intenses CNRS Grenoble, France
2 Outline Outline: Massless electrons in graphene and other solids (zoology) Optics and magneto-optics of electrons in conical bands Conclusions
3 Graphene 2D 2D crystal made of carbon atoms organized in hexagonal lattice Theoretically known over sixty years P. R. Wallace, Phys. Rev. 71, 622 (1947) Isolated/fabricated in 2004/2005 K. S. Novoselov et al., Science 306, 666 (2004) K. S. Novoselov et al., Nature 438, 197 (2005) Carbon Diamond Graphite nanotube 3D 3D 1D 0D Fulleren
4 Electronic band structure of graphene Crystal lattice: Electronic bands: Electronic dispersion (K points): Electrons in graphene = charged massless (relativistic) particles
5 Electronic band structure of graphene Effective Dirac Hamiltonian (for K or K point): Electronic bands: Electronic dispersion (K points): Electrons in graphene = charged massless (relativistic) particles
6 Highlights of graphene physics Half-integer and fractional QHE: Highlights of graphene physics K. Novoselov et al., Nature 438, 197 (2005) Y. Zhang et al., Nature 438, 201 (2005) K. I. Bolotin et al.,, Nature 462, 196 (2009) Room temperature QHE: K. Novoselov et al., Science 315, 1379 (2007) Universal dynamic conductivity: R. R. Nair et al., Science 320, 1308 (2008) A. B. Kuzmenko et al., Phys. Rev. Lett. 100, (2008)
7 Solids with relativistic-like electrons (timeline) 2005: Graphene & hexagonal systems systems K. S. Novoselov et al., Nature 438, 197 (2005) 2007: 2D topological insulators M. König et al., Science 318, 766 (2007) 2008: 3D topological insulators D. Hsieh et al., Nature 452, 970 (2008) 2011: Rashba-type semiconductors K. Ishizaka et al., Nature Mater. 10, 521 (2011) 2012: 3D topological crystalline insulators P. Dziawa et al., Nature Mater. 11, 1023 (2012) 2014: 3D Dirac semimetals Z. K. Liu et al., Science 343, 864 (2014) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) 2015: 3D Weyl semimetals B. Q. Lv et al., Nature Phys. 11, 724 (2015) 2016: Nodal-loop Dirac semimetals L. M. Schoop et al., Nature Comm. 7, (2016)
8 Solids with relativistic-like electrons (classification) Topological band theory: Band structures classified in terms of Z 2 invariant, Chern number, Berry phase, time-reversal/point/space symmetries see, e.g., A. Bansil et al., Rev. Mod. Phys. 88, (2016) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010)
9 Solids with relativistic-like electrons (classification) Topological band theory: Band structures classified in terms of Z 2 invariant, Chern number, Berry phase, time-reversal/point/space symmetries see, e.g., A. Bansil et al., Rev. Mod. Phys. 88, (2016) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) Electrons in conical bands, which differ by dimensionality, valley and spin degeneracy, (an)isotropy, time-reversal/lattice symmetry, corresponding Hamiltonian
10 Solids with relativistic-like electrons (timeline) 2005: Graphene & hexagonal systems systems K. S. Novoselov et al., Nature 438, 197 (2005) 2007: 2D topological insulators M. König et al., Science 318, 766 (2007) 2008: 3D topological insulators D. Hsieh et al., Nature 452, 970 (2008) 2011: Rashba-type semiconductors K. Ishizaka et al., Nature Mater. 10, 521 (2011) 2012: 3D topological crystalline insulators P. Dziawa et al., Nature Mater. 11, 1023 (2012) 2014: 3D Dirac semimetals Z. K. Liu et al., Science 343, 864 (2014) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) 2015: 3D Weyl semimetals B. Q. Lv et al., Nature Phys. 11, 724 (2015) 2016: Nodal-loop Dirac semimetals L. M. Schoop et al., Nature Comm. 7, (2016)
11 STM constant potential image Artificial graphene = C.-H. Park et al., Phys. Rev. Lett. 101, (2008) M. Gibertini et al., Phys. Rev. B 79, (2009) L. Nádvorník et al., New J. Phys. 14, (2012) Molecular graphene: Direct analogues of graphene hexagonally patterned 2D electron gas in a semiconductor QW STS spectrum Cu (111) surface with CO atoms assembled into hexagonal using STM/STS tip K. K. Gomes et. al, Nature 483, 306 (2012) Silicene, germanene, phosphorene S. Cahangirov et al., Phys. Rev. Lett. 102, (2009) H. Liu et al., ACS Nano 8, 4033 (2014)
12 Symmetry breaking in graphene-like systems Breaking hexagonal symmetry opens the energy band gap Cross-over from massless to massive Dirac electrons K. Novoselov, Nature Mater. 6, 720 (2007) Materials for valleytronics (e.g.): monolayers of h-bn or transitions metal dichalcogenides (MoS 2, WSe 2, MoSe 2 ) W. Yao et al., Phys. Rev. B 77, (2008)
13 Solids with relativistic-like electrons (timeline) 2005: Graphene & hexagonal systems systems K. S. Novoselov et al., Nature 438, 197 (2005) 2007: 2D topological insulators M. König et al., Science 318, 766 (2007) 2008: 3D topological insulators D. Hsieh et al., Nature 452, 970 (2008) 2011: Rashba-type semiconductors K. Ishizaka et al., Nature Mater. 10, 521 (2011) 2012: 3D topological crystalline insulators P. Dziawa et al., Nature Mater. 11, 1023 (2012) 2014: 3D Dirac semimetals Z. K. Liu et al., Science 343, 864 (2014) M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 82, 3045 (2010) 2015: 3D Weyl semimetals B. Q. Lv et al., Nature Phys. 11, 724 (2015) 2016: Nodal-loop Dirac semimetals L. M. Schoop et al., Nature Comm. 7, (2016)
14 Insulating in bulk, but conducting via conical band(s) on the surface Topological insulators Effective Hamiltonian for surface states (Rashba-like): The conical band protected by time-reversal symmetry 2D and 3D topological insulators: (3D/2D bulk + 1D/2D surface) HgTe/HgCdTe and InAs/GaSb QWs Bi 1-x Sb x, Bi 2 Se 3, Bi 2 Te 3, Sb 2 Te 3 M. König et al., Science 318, 766 (2007) I. Knez et al., Phys. Rev. Lett. 107, (2011) D. Hsieh et al., Nature 452, 970 (2008) H. Zhang et al., Nature. Phys. 9, 438 (2009) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, (2005) B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, (2006) M. Z. Hasan et al., Rev. Mod. Phys. 82, 3045 (2010)
15 Topological insulators Insulating in bulk, but conducting via conical band(s) on the surface Surface states of Bi 2 Se 3 in ARPES: Effective Hamiltonian for surface states (Rashba-like): The conical band protected by time-reversal symmetry 2D and 3D topological insulators: (3D/2D bulk + 1D/2D surface) credit to Y. L. Chen, Oxford HgTe/HgCdTe and InAs/GaSb QWs Bi 1-x Sb x, Bi 2 Se 3, Bi 2 Te 3, Sb 2 Te 3 M. König et al., Science 318, 766 (2007) I. Knez et al., Phys. Rev. Lett. 107, (2011) D. Hsieh et al., Nature 452, 970 (2008) H. Zhang et al., Nature. Phys. 9, 438 (2009) C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, (2005) B. A. Bernevig and S.-C. Zhang, Phys. Rev. Lett. 96, (2006) M. Z. Hasan et al., Rev. Mod. Phys. 82, 3045 (2010)
16 Insulating in bulk, but conducting via conical band(s) on the surface Topological crystalline insulators Effective Hamiltonian for surface states (Rashba-like): The conical band protected by crystalline symmetry e.g., Pb 1-x Sn x Se, Pb 1-x Sn x Te, SnTe L. Fu et al., Phys Rev. Lett. 106, (2011) P. Dziawa et al., Nature Mater. 11, 1023 (2012)
17 3D Dirac semimetals Bulk systems with one or more spin-degenerate 3D conical bands Na 3 Bi, Cd 3 As 2, gapless HgCdTe, ZrTe 5 Z. K Liu et al., Science 343, 864 (2014) Z. K. Liu et al., Nature Mater. 13, 677 (2014) S. Jeon et al., Nature Mater. 13, 851 (2014) M. Orlita et al., Nature Phys. 10, 233 (2014) S. Borisenko et al., Phys. Rev. Lett. 113, (2014) M. Neupane et al., Nature Comm. 5, 3786 (2014) R. Y. Chen et al., Phys. Rev. Lett. 115, (2015) Effective Hamiltonian (3D Weyl + spin degeneracy): Conical bands may be protected by the lattice symmetry B.-J. Yang & N. Nagaosa, Nature Comm.5, 4898 (2014)
18 3D Weyl semimetals 3D conical bands with no spin degeneracy = with either time- or inversion-symmetry lifted TaAs, NbAs, TaP B. Q. Lv et al., Nature Phys. 11, 724 (2015) L. X. Yang et al., Nature Phys. 11, 728 (2015) S.-Y. Xu et al., Nature Phys. 11, 748 (2015) Effective (Weyl) Hamiltonian: Weyl cones always in pairs! Fermi arc = specific, in k-space disjoint surface state
19 Valley degenerated Spin degenerated Conical bands in solids: Overview & examples Dimensionality 1D 2D 3D HgTe QW (critical thickness) Büttner et al., Nature Phys D Dirac semimetals (w/o symmetry protection) (gapless HgCdTe, ZrTe 5 ) Orlita Nature Phys Chen et al., PRL 2015 Metallic carbon nanotubes see, e.g. Ando, SST D topological insulators inverted HgTe QWs König et al., Science 2007 InAs/GaSb QWs Knez et al., PRL 2011 Graphene Novoselov et al., Nature D topological insulators (Bi 1-x Sb x, Bi 2 Se 3, Bi 2 Te 3 ) Hsieh et al., Nature 2008 Rashba-type semiconductors (BiTeI, BiTeCl, BiTeBr) Ishizaka et al., Nature. Mater D Dirac semimetals (Cd 3 As 2, Na 3 Bi) Liu et al., Science 2014 Liu et al, Nature Mater D Weyl semimetals (e.g., TaAs, NdAs, TaP) Lv et al., Nature Phys Xu et al., Nature Phys. 2015
20 Density of states: conventional systems
21 Density of states: conical bands
22 Optical response of electrons in conical bands v v 1 st order, electric-dipole excitations considered only
23 Conical bands: Absorption of light on free carriers Classical Drude model of (optical) conductivity v v
24 Conical bands: Absorption of light on free carriers Classical Drude model of (optical) conductivity Free-carrier absorption graphene v v M. Orlita et al., New. J. Phys. 14, (2012)
25 Conical bands: Absorption of light on free carriers Classical Drude model of (optical) conductivity Free-carrier absorption graphene v v M. Orlita et al., New. J. Phys. 14, (2012) dc conductivity
26 Conical bands: Absorption of light on free carriers Classical Drude model of (optical) conductivity Free-carrier absorption graphene v v dc conductivity Drude (optical) weight
27 Interband excitations in conical bands v Optical band gap (zero T, finite doping) Absorption of light in solids (Fermi s golden rule & electric dipole excitations): Joint density of states: v
28 Interband excitations in conical bands R. R. Nair et al., Science 320, 1308 (2008) Absorption coefficient: For conical bands in 2D: Dispersionless interband absorption of light
29 Interband excitations in conical bands R. R. Nair et al., Science 320, 1308 (2008) Absorption coefficient: For conical bands in 2D: Dispersionless and universal interband absorption of light
30 Interband excitations in conical bands Flat absorption of light (2.3%) defined only by the fine structure constant: R. R. Nair et al., Science 320, 1308 (2008) A. B. Kuzmenko et al., Phys. Rev. Lett. 100, (2008)
31 Interband excitations in conical bands v Optical band gap (zero T, finite doping) Absorption of light in solids (Fermi s golden rule & electric dipole excitations): Joint density of states: v
32 3D conical band: optical conductivity Gapless HgCdTe: Absorption of light in solids (e.g., Fermi s golden rule): For conical bands in 3D: Absorption coefficient linear in photon frequency! M. Orlita et al., Nature Phys. 10, 233 (2014)
33 3D conical band: optical conductivity Cadmium arsenide (Cd 3 As 2 ): Absorption of light in solids (e.g., Fermi s golden rule): For conical bands in 3D: Absorption coefficient linear in photon frequency! A. Akrap et al., Phys. Rev. Lett. 117, (2016)
34 3D conical band: optical conductivity Cadmium arsenide (Cd 3 As 2 ): Absorption of light in solids (e.g., Fermi s golden rule): For conical bands in 3D: Absorption coefficient linear in photon frequency! Cut off due to Pauli blocking (=> E F )
35 Optical conductivity of a conical band: summary v (Joint) density of states: Optical conductivity: v
36 Optical response of electrons in conical bands v v 1 st order, electric-dipole excitations considered only
37 Optical response of electrons in conical bands v Magnetic field? v 1 st order, electric-dipole excitations considered only
38 Cyclotron resonance massive electrons Charged particle in magnetic field: Cyclotron motion at the frequency: Cyclotron resonance = resonant absorption of light at the cyclotron frequency
39 Cyclotron resonance in solid-state physics Germanium = the first solid-state system in which cyclotron resonance was observed G. Dresselhaus, A. F. Kip, and C. Kittel Phys. Rev. 92, 827 (1953) More than the estimate of the effective mass, important observation for the concept of quasi-particles in condensed matter physics see, e.g., M. L. Cohen, AIP Conference Proceedings 772, 3 (2005)
40 Cyclotron resonance Charged particle in magnetic field: Cyclotron motion at the frequency: Cyclotron resonance = resonant absorption of light at the cyclotron frequency
41 Cyclotron motion of massless electrons (classical description) Charged particle in magnetic field: Cyclotron motion at the frequency: Linear in B Cyclotron mass (energy dependent) "Effective" effective mass of massless particles, i.e., Einstein energy-mass relation General definition of cyclotron mass: see, e.g., Ashcroft & Mermin
42 Conical bands: Absorption of light on free carriers Classical Drude model for optical conductivity Free-carrier absorption graphene v v dc conductivity Drude (optical) weight
43 Cyclotron motion of massless electrons (classical description) Charged particle in magnetic field: Cyclotron motion at the frequency: Linear in B Cyclotron mass (energy dependent) "Effective" effective mass of massless particles, i.e., Einstein energy-mass relation General definition of cyclotron mass: see, e.g., Ashcroft & Mermin
44 Cyclotron resonance in graphene Quasi-free-standing graphene on SiC in classical regime Magneto-absorbance M. Orlita et al., New J. Phys. 14, (2012) A. M. Witowski et al., Phys. Rev. B 82, (2010)
45 Cyclotron resonance in graphene Landau levels: Quantum regime Selection rules: Cyclotron resonance Interband excitations
46 Landau level spectrum & optical excitations
47 Interband inter-landau level excitations Cyclotron resonance in graphene Cyclotron resonance Magneto-absorbance Multilayer epitaxial graphene on SiC in quantum regime M. Orlita et al., Phys. Rev. Lett. 101, (2008)
48 Interband inter-landau level excitations Cyclotron resonance in graphene Cyclotron resonance Magneto-absorbance Energy spectrum: Velocity parameter: M. Orlita et al., Phys. Rev. Lett. 101, (2008)
49 Interband inter-landau level excitations Cyclotron resonance in graphene Cyclotron resonance Magneto-absorbance Carrier mobility: M. Orlita et al., Phys. Rev. Lett. 101, (2008)
50 Cyclotron resonance in 3D Dirac semimetal Cd 3 As 2 High-field magneto-reflectivity of Cd 3 As 2 : Velocity, anisotropy of conical bands A. Akrap et al., Phys. Rev. Lett. 117, (2016) M. Hakl et al., Phys. Rev. B 97, (2018)
51 Landau levels spectroscopy of 3D Dirac electrons in gapless ZrTe 5 Magneto-transmission: G Fan chart: Gap, velocity, Zeeman splitting (g factors) I. Crassee, A. Akrap et al., unpublished (2018) see also, Z. G. Chen et al., PNAS 114, 816 (2017) & R. Y. Chen et al., Phys. Rev. Lett. 115, (2015)
52 credit to QPEC, University of Tokyo Landau level spectroscopy of BiTeI Rashba-type spin splitting of conduction-band electrons: Bulk electrons (2D massless + 1D massive) Magneto-transmission: Fan chart: Velocity, departure from (= from band linearity) S. Bordacs et al., Phys. Rev. Lett. 111, (2013) S. Bordacs et al., unpublished (2017)
53 Universal magneto-optical effects in topological insulators Magneto-optics on a thin layer of a topological insulator: M. Z. Hasan, Physics 3, 62 (2010) Universal Faraday/Kerr rotations (determined by fine structure constant a only) predicted for topological insulators with broken TR-symmetry W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett. 105, (2010) J. Maciejko et al., Phys. Rev. Lett. 105, (2010)
54 Universal magneto-optical effects in topological insulators Tested in very first experimental studies. L. Wu et al., Science 354, 1124 (2016) V. Dziom et al., Nature Comm. 8, (2017) Kerr rotation on surface states of Bi 2 Se 3 : Plateaus in the Kerr/Faraday angle defined by the fine structure only
55 Relativistic quantum electrodynamics in 3D Weyl semimetals? Chiral anomaly = generation/annihilation of chiral fermions from/to vacuum ( ) S. L. Adler, Phys. Rev. 177, 2426 (1969) Solid-state analog = transfer of electrons between Weyl nodes H. B. Nielsen, Phys. Lett. 130B, 389 (1983) Negative magneto-resistance not a unique proof magneto-optical studies? R. D. dos Reis et al., New Journal of Phys. 18, (2016) P. E. C. Ashby and J. P. Carbotte, Rev. B 89, (2014)
56 Relativistic quantum electrodynamics in 3D Weyl semimetals? Chiral anomaly = generation/annihilation of chiral fermions from/to vacuum ( ) S. L. Adler, Phys. Rev. 177, 2426 (1969) Solid-state analog = transfer of electrons between Weyl nodes H. B. Nielsen, Phys. Lett. 130B, 389 (1983) Chiral anomaly via splitting of interband absorption edge:
57 External collaborators - acknowledgement Optical spectroscopy: M. Hakl, C. Faugeras, G. Martinez, M. Potemski LNCMI, CNRS, Grenoble, France A. Akrap, I. Crassee, D. van der Marel Université de Genève, Switzerland C. C. Homes Brookhaven National Laboratory, USA J. Kuba CEITEC & BUT, Brno, Czech Republic L. Wu, N. P. Armitage Johns Hopkins University, Baltimore, USA S. Bordacs Budapest University, Hungary Theory: S. Tchoumakov, M. O. Goerbig LPS, CNRS, Paris Orsay, France C. Michel, E. M. Hankiewicz Wűrzburg University, Germany X-ray: O. Caha, J. Novák S. Koohpayeh CEITEC & Masaryk University, Brno, Czech Republic Johns Hopkins University, Baltimore, USA Sample growth: A. Arushanov, A. Nateprov Institute of Applied Physics, ASM, Moldova Q. D. Gibson, R. J. Cava Princeton University, USA W.-L. Lee, R. Sankar Academia Sinica, Tai-pei, Taiwan C. Gould, C. Brune, L. Molenkamp Wűrzburg University, Germany C. Berger, W. A. de Heer GeorgiqTech, Atlanta, USA T. Seyller TU Chemnitz, Germany Transport: B. A. Piot LNCMI, CNRS, Grenoble, France F. Teppe, W. Desrat LCC, CNRS & Université Montpellier, France
58 Conclusions Conclusions/Summary Since discovery/fabrication of graphene triggered a search for other systems with conical bands (in bulk and/or on their surfaces) : Topological insulators, topological crystalline insulators, 3D Dirac and Weyl semimetals, Rashba-type semiconductors Optical and magneto-optical spectroscopy (in the THz and infrared) is a well-suited experimental method to explore them: Band structure parameters masses, velocities, gaps; carrier density; scattering mechanisms, relaxation times/mobilities; phenomena due to electron-phonon or electron-electron interaction; appealing universal and QED effects
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