Floquet topological insulator in semiconductor quantum wells

Journal name:
Nature Physics
Year published:
Published online


Topological phases of matter have captured our imagination over the past few years, with tantalizing properties such as robust edge modes and exotic non-Abelian excitations, and potential applications ranging from semiconductor spintronics to topological quantum computation. Despite recent advancements in the field, our ability to control topological transitions remains limited, and usually requires changing material or structural properties. We show, using Floquet theory, that a topological state can be induced in a semiconductor quantum well, initially in the trivial phase. This can be achieved by irradiation with microwave frequencies, without changing the well structure, closing the gap and crossing the phase transition. We show that the quasi-energy spectrum exhibits a single pair of helical edge states. We discuss the necessary experimental parameters for our proposal. This proposal provides an example and a proof of principle of a new non-equilibrium topological state, the Floquet topological insulator, introduced in this paper.

At a glance


  1. Inducing an FTI from a trivial insulator.
    Figure 1: Inducing an FTI from a trivial insulator.

    Energy dispersion ε(k) and pseudospin configuration for the original bands of (k) in the non-topological phase (M/B<0). The non-topological phase is characterized by a spin-texture that does not wrap around the unit sphere. On application of a periodic modulation of frequency ω greater than the bandgap, a resonance appears; the green circles and arrow depict the resonance condition.

  2. A topological Floquet band.
    Figure 2: A topological Floquet band.

    Pseudospin configuration (blue arrows) and dispersion of the lower band of HI. Note the dip in the energy surface near k=0, resulting from the reshuffling of the lower and upper bands of (k).

  3. The geometrical condition for creating topological quasi-energy bands.
    Figure 3: The geometrical condition for creating topological quasi-energy bands.

    The purple arrow and green circle depict on the curve γ in the FBZ (depicted on the right), for which the resonant condition holds. The red arrow and curve depict on γ. The blue arrow depicts the driving field vector V. As long as V points within the loop traced by , the vector winds around the north pole, which is indicated by the black arrow.

  4. Edge states in the quasi-energy spectrum.
    Figure 4: Edge states in the quasi-energy spectrum.

    Quasi-energy spectrum of the Floquet equation (3) of the Hamiltonian (17), in the strip geometry: periodic boundary conditions in the x direction, and vanishing ones in the y direction. The driving field was taken to be in the direction. The horizontal axis labels the momentum kx. The vertical axis labels the quasi-energies in units of |M|. Two linearly dispersing chiral edge modes traverse the gap in the quasi-energy spectrum. The parameters used are ω=2.3|M|, |V|=A=|B|=0.2|M|. The inset shows the dispersion of the original Hamiltonian (17), for the same parameters.

  5. Time dependence of the edge states.
    Figure 5: Time dependence of the edge states.

    Density of edge mode as function of time, |ϕ(y,t)|2, a for kx=0, and b for kx=0.84, where the edge modes meet the bulk states. The horizontal axis shows the distance from the edge, y, in units of the lattice constant, and the time in units of 2π/ω. For clarity the density for only the 20 lattice sites closest to the edge are shown.


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  1. Institute of Quantum Information, California Institute of Technology, Pasadena, California 91125, USA

    • Netanel H. Lindner &
    • Gil Refael
  2. Department of Physics, California Institute of Technology, Pasadena, California 91125, USA

    • Netanel H. Lindner &
    • Gil Refael
  3. Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

    • Victor Galitski
  4. Joint Quantum Institute, Department of Physics, University of Maryland, College Park, Maryland 20742, USA

    • Victor Galitski


N.H.L., G.R. and V.G. contributed to the conceptual developments. N.H.L. carried out the mathematical analysis.

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