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High-order multipole radiation from quantum Hall states in Dirac materials

Michael J. Gullans, Jacob M. Taylor, Ataç Imamoğlu, Pouyan Ghaemi, and Mohammad Hafezi
Phys. Rev. B 95, 235439 – Published 30 June 2017
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  • INTRODUCTION
  • DIRAC MODEL
  • RADIATION FROM THE EDGE
  • RADIATION FROM THE BULK
  • ELECTRON-ELECTRON INTERACTIONS
  • CONCLUSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    We investigate the optical response of strongly disordered quantum Hall states in two-dimensional Dirac materials and find qualitatively different effects in the radiation properties of the bulk versus the edge. We show that the far-field radiation from the edge is characterized by large multipole moments (>50) due to the efficient transfer of angular momentum from the electrons into the scattered light. The maximum multipole transition moment is a direct measure of the coherence length of the edge states. Accessing these multipole transitions would provide new tools for optical spectroscopy and control of quantum Hall edge states. On the other hand, the far-field radiation from the bulk appears as random dipole emission with spectral properties that vary with the local disorder potential. We determine the conditions under which this bulk radiation can be used to image the disorder landscape. Such optical measurements can probe submicron-length scales over large areas and provide complementary information to scanning probe techniques. Spatially resolving this bulk radiation would serve as a novel probe of the percolation transition near half filling.

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    • Received 26 January 2017

    DOI:https://doi.org/10.1103/PhysRevB.95.235439

    ©2017 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Optical & microwave phenomenaQuantum Hall effect
    1. Physical Systems
    2-dimensional systems
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Michael J. Gullans1, Jacob M. Taylor1, Ataç Imamoğlu2, Pouyan Ghaemi3,4, and Mohammad Hafezi5

    • 1Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, NIST and University of Maryland, College Park, Maryland 20742, USA
    • 2Institute of Quantum Electronics, ETH Zürich, CH-8093 Zürich, Switzerland
    • 3Physics Department, City College of the City University of New York, New York, New York 10031, USA
    • 4Physics Department, Graduate Center, City University of New York, New York, New York 10031, USA
    • 5Department of Electrical and Computer Engineering, IREAP, and Joint Quantum Institute, University of Maryland, College Park, Maryland 20742, USA

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    Issue

    Vol. 95, Iss. 23 — 15 June 2017

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    Images

    • Figure 1
      Figure 1

      (a) In the presence of a large magnetic field, the electronic states of the 2DDM are quantized into Landau levels, which we index by their angular momentum −ℏm. The majority of the states in the bulk are localized by disorder, leading to interband radiation dominated by dipole emission. The spectrum of this radiation is spatially correlated with the disorder potential. Here, Ec(v) refer to the energy of the conduction (valence) band and EF is the Fermi energy. (b) An electron excited at the edge of the system can emit light with orbital angular momentum ℏℓ by recombining with a hole in the state, m′=m−ℓ.

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    • Figure 2
      Figure 2

      (a) Low-energy band structure of graphenelike Dirac material for zero magnetic field. Here, m0 and v are the Dirac mass and velocity, respectively, and we only show one of the two valleys. (b) Amplitude of the cylindrical vector harmonic |Eℓ| for ℓ=100 with λ0=600nm and index of refraction n0=3.2. Because the size of the optical vortex increases as λ−ℓ, an edge state with radius re (black circle) can only spontaneously emit into modes with ℓ≲re/λ−. (c) Branching ratio for spontaneous emission into different ℓ modes for two different values of re/λ−. We took Dirac parameters for WSe2 (m0v2≈1 eV and v≈106 m/s [51]) embedded in GaP, Bz=11 T, n=0, and λ−=30 nm.

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    • Figure 3
      Figure 3

      (a) The disorder potential U(x,y) for the interband transitions between Landau levels. (b) U(x,y) can be reconstructed by correlating the amplitude of spatially resolved scattered light with the frequency of the incoming probe. We took the 2DDM to be embedded in GaP (n0=3.2) in a 10 T magnetic field with λ0=1μm. The optical imaging is able to resolve spatial features down to the diffraction limit λ0/2n0≈160 nm.

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    • Figure 4
      Figure 4

      Scaling of multipole emission rate γℓ with increasing orbital angular momentum quantum number ℓ in the regime where the dipole approximation breaks down, k0rm≫1.

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