Nonlinear Optical Response of Massless Dirac Fermions in Graphene and Topological Materials

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1 Nonlinear Optical Response of Massless Dirac Fermions in Graphene and Topological Materials Alexey Belyanin Department of Physics and Astronomy Texas A&M University Zhongqu Long, Sultan Almutairi, Ryan Kutayah, and Yongrui Wang Texas A&M University Collaborations: Mikhail Tokman, Russian Academy of Sciences Manfred Helm, Dresden

2 Papers on the project published since last review Y. Wang, M. Tokman, and A. Belyanin, Second-order nonlinear optical response of graphene, Phys. Rev. B 94, (2016). Exact analytic formulas for strong degeneracy Satisfy all permutation and symmetry properties, gauge invariance, and semiclassical limit J. König-Otto, Y. Wang, A. Belyanin, C. Berger, W. de Heer, M. Orlita, A. Pashkin, H. Schneider, M. Helm, S. Winnerl, Four- Wave Mixing in Landau-Quantized Graphene, Nano Letters, in press Q. Zhang, Y. Wang, W. Gao, J. D. Watson, M. J. Manfra, A. Belyanin, and J. Kono, Stability of high-density twodimensional excitons against a Mott transition in high magnetic fields probed by coherent terahertz spectroscopy, Phys. Rev. Lett. 117, (2016).

3 New research in 2017 Linear and nonlinear optics of massless chiral fermions in Dirac and Weyl semimetals Electrodynamics of 2D materials in nanocavities: Full QED theory of light-matter interaction going beyond conventional Fermi Golden rule and Purcell factor; Enhancement of spontaneous emission and absorption Enhancement of nonlinear wave mixing and parametric decay Landau-Zener transitions and ballistic breakdown of graphene by a strong THz pulse Superfluorescence in thin films of single-walled carbon nanotubes

4 Massless Dirac fermions Graphene Surface states in 3D topological insulators 3D Dirac fermions in Dirac semimetals 1D chiral fermions in Weyl semimetals Fascinating optics and electrodynamics!

5 Band structure of graphene Linear dependence and conical selfcrossing for E(p) at E = 0 in K,K points Fermi energy is at E = 0 in undoped graphene Low-energy excitations are massless chiral Dirac fermions! This includes next-to-nearest neighbors, which lightly shifts the conical points and makes the cones asymmetric Geim, Phys. Today 2007 Castro Neto et al. RMP 2009

6 Band structure near K- or K -point One can assign pseudospin ±½ and write Hamiltonian in terms of Pauli matrices. It has nothing to do with real spin. velocity Eigenstates: spinors Energy æ s x = 0 1 ö æ ç,s è 1 0 y = 0 -i ö ç ø è i 0 ø Chirality: definite pseudospin direction (parallel or antiparallel to p)

7 3D topological insulators: Band insulator in the bulk, 2D gapless metallic states on the surface Suppressed backward scattering Bi 2 Se 3, Sb 2 Te 3 : large bulk band gap ~ 0.3 ev jarilloherrero.mit.edu wikipedia Xia et al. 2009

8 Graphene vs. topological insulators Graphene: pseudospin ±½ related to A,B sublattices; Spin and valley degeneracy Topological insulator: real spin; no degeneracy u F»10 8 cm/s; u' F» cm/s No valley and spin degeneracy One spin direction per surface state Spin locked to the direction of momentum k and rotates with k. No two states at same energy and spin Momentum scattering is suppressed

9 3D massless Dirac fermions Discovery of 3D Dirac/Weyl semimetal Na 3 Bi: Liu et al., Science 2014 Dirac Hamiltonian (4X4): H = c(a x p x +a y p y +a z p z )+ bmc 2

10 Weyl semimetal: need to violate inversion or time-reversal symmetry Hasan, Science 2015: discovery of Weyl fermion semimetal TaAs

11 Applying a strong magnetic field to Dirac semimetal Na 3 Bi converts it into Weyl semimetal Weyl Hamiltonian (2X2): H = ±u F (s x p x +s y p y +s z p z ) Topologically protected Dirac points From Xiong et al., Science 2016

12 1D chiral fermions in Weyl semimetals Chirality, or Chern number, or flux of Berry curvature through Fermi surface enclosing the Weyl node

13 Chiral anomaly We can derive it from plasma response in the limit q = 0 (uniform field) Hosur & Qi, 2013

14 Energy, mev B = 1 Tesla Energy, mev E F k z, 10 6 cm -1 k z, 10 6 cm -1 x y θ q z B E y ordinary wave E x, E z extraordinary wave

15 Dielectric response of n = 0 particles: 1D chiral plasma! Energy, mev E F Response to the field along z k z, 10 6 cm -1 Quantum-mechanical density-matrix theory for longitudinal current

16 Energy, mev Longitudinal dielectric function: Effective plasma frequency: Gives chiral anomaly when q z = Does not depend on electron density! Depends only on the magnetic field W n=1 -W n= B, Tesla

17 Transverse dielectric response: inter-landau-level transitions Energy, mev E F Δ n = +1 LHC Δ n = -1 RHC k z, 10 6 cm -1 Plot for B = 10 Tesla

18 Absorption for LHC and RHC light: strong polarization effects! Onset of 0 -> 1 LHC transition -1 -> 0-2 -> 1 RHC -1 -> 2 LHC

19 Energy, mev Energy, mev E k F z, 10 6 cm -1 k z, 10 6 cm -1 E F Polarization effects due to chiral anomaly Consider for simplicity a high magnetic field 10 T, low temperature, scattering rate of 5 mev, and Fermi energies 100 mev in one node and -100 mev in another node

20 Optical polarization detection of chiral anomaly E F = 100 mev in both nodes E F = 100 mev in one node and mev in another node Both polarizations show the same absorption!

21 Energy, mev Energy, mev k z, 10 6 cm -1 k z, 10 6 cm -1 x y θ q z B E y ordinary wave E x, E z extraordinary wave

22 Electromagnetic wave propagation in Weyl semimetals Note that ε zz can be negative at low frequencies! Dispersion equation: x y θ q z B If all components of the dielectric tensor are positive: ellipsoid If transverse and zz components are of opposite sign: hyperbolic surfaces Natural hyperbolic material!

23 Kivshar, Nat. Phot Hyperbolic metamaterials

24 Electromagnetic wave propagation in Weyl semimetals Frequencies below the Landau-level absorption edge; Fermi energy at the Dirac point Ordinary wave, E y: Extraordinary wave, E x, E z Propagation transverse to B (θ = π/2): x Propagation parallel to B (θ = 0): splits into photon + plasmon q Photon: y θ z B Plasmon:

25 No Landau damping for 1D chiral fermions! Energy, mev k z, 10 6 cm -1 Velocity does not depend on momentum; bunching is impossible! No poles at v = ω/q in the integration over momenta; no contribution from residues

26 Hybrid modes in Weyl semimetals: oblique propagation Coupling of transverse photons and longitudinal plasmons! Rotate the coordinates to have z axis parallel to wave vector q Dispersion equation: Extraordinary wave dispersion: Hybrid resonance at Extremely rich dispersion; possibility to excite hybrid modes from vacuum at n = 1

27 Enhancement of spontaneous emission and parametric processes for 2D materials in nanocavities Mikkelsen, Nat. Comm Typical enhancement of the parametric signal as compared to a Fabry-Perot cavity: æ µ L ö z ç è l ø 2 æ ç è l ö ø L FP 2 æ ç è Dw FP Dw eff ö ø 2

28 Modeling: classical pumping, quantized signal and idler fields w p =w s +w i Operator-valued Maxwell equations Couples modes, leads to entanglement and parametric instability Overlap of three cavity modes: replaces phase-matching condition J = ò z ( p x, y)z ( i x, y)z ( s x,y)d 2 r S Then add dissipation and fluctuation terms (Heisenberg-Langevin approach)

29 Landau-Zener transitions and ballistic breakdown of graphene by a strong THz pulse THz pulses with the electric field kv/cm are available Electron energy in a field is much larger than the optical transition energy P LZ» e -2pG

30 V s s k k = k,s (-ej ˆ1 ) k, s = ee t Density-matrix theory æ ( ) ç è sinq ( k) d k 2k -i 1+ s s k 2 d( k-x0 q) lim -d k k k q 0 q ö ø Note the convective terms µ k x

31 Population inversion right after a 100-fs 5-THz pulse

32 Interaction Hamiltonian defined by a vector potential V s s k k = k,s e c u ˆ eu Fs x A k, s = d F A t k k c ( ) ( seiq k) + s e -iq ( k ) 2 Density-matrix equations n S t = 0

33 Generation of harmonics by several-cycle pulses

34 Back to graphene: Landau levels in graphene vs. quantum wells e n µ n B e n µ ç n + 1 è 2 æ ö B ø E(p) = u F p E(p) = p2 2m p B B

35 Massless Dirac fermions in a magnetic field Quantum-mechanical picture N = A 2pl c 2 n = 2 n = 1 n = 0 n = -1 Leggett lectures B B

36 Frequencies in the mid-ir to THz range in a magnetic field of a few Tesla ±1 0 ±1 1 2

37 Enhanced optical nonlinearity No magnetic field Magnetic field Inter-Landau Level transition 4-wave-mixing from interband transitions c (3) ~ 10-7 esu In the near-ir Hendry et al. PRL 105,097401, 2010 Enhanced density of states ~ w c g Single- and multi-photon resonance Much stronger optical nonlinearity Yao & Belyanin, PRL 108, (2012) PRL 110, (2013)

38 Resonant 3 rd order nonlinearity P (3) (w 3 ) = c (3) (w 3 )E 1 (w 1 )( E * 2 (w 2 )) 2 w 3 = w 1-2w 2 w 1 w 2 w 2 Assuming the same scattering rates γ Divided by thickness Bulk susceptibility For pump below saturation, f(ρ) ~ 1 and χ (3) 0.4 esu at B = 1 T. Scales as 1/B Compare with χ (3) 10-7 esu for asymmetric quantum wells

39 First 4-wave mixing experiment Collaboration with M. Helm group (Dresden). Nano Lett. in press

40 Outlook New materials with massless Dirac/Weyl fermions: exciting optical properties Natural hyperbolic materials No Landau damping Plasmon resonance and hybrid modes Optical diagnostics of chiral anomaly Combining 2D materials with nanocavities: Heisenberg-Langevin theory Enhancement of light-matter interactions Strong-field THz pulses: Landau-Zener-Schwinger-Klein physics Putting four-wave mixing theory for Landau-quantized graphene to the test