Research ArticleCONDENSED MATTER PHYSICS

Quantum spin Hall insulator with a large bandgap, Dirac fermions, and bilayer graphene analog

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Science Advances  20 Apr 2018:
Vol. 4, no. 4, eaap7529
DOI: 10.1126/sciadv.aap7529
  • Fig. 1 Band structure of three-layer InAs/GaSb QWs with InAs geometry.

    (A) Schematic representation of symmetrical three-layer InAs/GaSb QWs with InAs geometry. The numbers show the bandgap values in materials of the layers. Here, d1 and d2 are the thicknesses of InAs and GaSb layers, respectively. The QW is supposed to be grown on (001) GaSb buffer. (B) Energy of electron-like (blue curves) and heavy hole–like (red line) subbands at k = 0, as a function of d1 at d2 = 4 nm. Zero energy corresponds to the top of the valence band in bulk GaSb. (C) Phase diagram for different d1 and d2. The white open region is a BI phase, whereas the gray-striped region defines an SM phase. The gray open region corresponds to the QSHI state. (D to F) 3D plot of the band structure at BI (D), QSHI (E), and SM (F) phases. The x and y axes are oriented along (100) and (010) crystallographic directions, respectively. The thicknesses of the layers for each phase, used in the calculations, are marked in (C) by blue open symbols.

  • Fig. 2 QSHI in three-layer InAs/GaSb QWs with InAs geometry.

    (A) Bandgap in the QSHI state as a function of d1 in the InAs/GaSb QWs with InAs geometry at d2 = 14 MLs, where one ML = 0.5a0 (a0 is a lattice constant in the bulk material). The black and red curves correspond to the QWs grown on InAs and GaSb buffers, respectively. The inset shows the maximum gap, which can be achieved at a given value of d2. (B) Bandgap in the QSHI state at different values of d1 and d2 = 14 MLs as a function of strain InAs layers ε1 and GaSb layer ε2. Three specific cases for the values of ε1 and ε2, corresponding to the InAs, GaSb, and AlSb buffers, are marked by vertical dotted lines. (C) Band dispersion in the QSHI state calculated on the basis of HInAs(kx, ky), with the layer thicknesses marked in Fig. 1C by blue open symbols. Electron- and heavy hole–like subbands are shown in blue and red, respectively. The black curves correspond to the dispersion of the edge states, obtained by numerical diagonalization of HInAs(kx, ky) with open boundary conditions along the y axis. Different Kramers partners are shown by solid and dotted curves. (D) LLs for the QW grown on GaSb buffer at d1 =32 MLs and d2 = 14 MLs. The numbers over the curves correspond to the LL indices (30). Two red curves are the zero-mode LLs, which are identified within an effective 4 × 4 Hamiltonian for E1 and H1 subbands (2). (E) Critical magnetic field Bc as a function of d1 with d2 = 14 MLs. The black and red curves correspond to the structures grown on InAs and GaSb buffers, respectively. The dotted vertical lines mark the values of d1SM.

  • Fig. 3 Band structure of three-layer InAs/GaSb QWs with GaSb geometry.

    (A) Schematic representation of symmetrical three-layer InAs/GaSb QWs with GaSb geometry. The numbers show the bandgap values in materials of the layers. Here, d1 and d2 are the thicknesses of InAs and GaSb layers, respectively. The QW is supposed to be grown on (001) GaSb buffer. (B) Energy of electron-like (blue curves) and heavy hole–like (red curves) subbands at k = 0, as a function of d2 at d1 = 10 nm. Zero energy corresponds to the top of valence band in bulk GaSb. (C) Phase diagram for different d1 and d2. The white open region is a BI phase, whereas the gray-striped region defines an SM phase. The gray open region corresponds to a BG state. (D to F) 3D plot of the band structure at BI (D), BG (E), and SM (F) phases. The x and y axes are oriented along (100) and (010) crystallographic directions, respectively. The thicknesses of the layers for each phase used in the calculations are marked in (C) by blue open symbols.

  • Fig. 4 BG phase in three-layer InAs/GaSb QWs with GaSb geometry.

    (A) Band dispersions for a BG phase in the QW grown on GaSb buffer at d1 = 10 nm and d2 = 6 nm. Electron- and hole-like subbands are shown in blue and red, respectively. Solid and dotted curves correspond to different spin states. (B) LL fan chart. The zero-mode LL, which has doubled degeneracy order as compared with other levels, is marked by red bold curve (the h1 and h2 levels). This LL is formed by states of both H1 and H2 subbands. LL, containing only the states from the E1 subband in high magnetic fields, is given in blue. The crossing between the e1 and h1 LLs, arising at critical magnetic field Bc ≈ 3.7 T, leads to the phase transition into normal (noninverted) band structure, as it is in single HgTe QW (11). (C) Band dispersions in an electric field of 5 kV/cm oriented perpendicular the QW plane. (D) Bandgap as a function of an applied electric field. (E) Band dispersion for a BG phase, calculated on the basis of HGaSb(kx, ky) with the layer thicknesses marked in Fig. 3C by blue open symbol. The black curves correspond to the dispersion of the edge states, obtained by numerical diagonalization of HGaSb(kx, ky) with open boundary conditions along the y axis. Different Kramers partners are shown by solid and dotted curves.

  • Fig. 5 Massless Dirac fermions in three-layer InAs/GaSb QWs.

    (A and B) 3D plot of the band dispersions of three-layer InAs/GaSb QWs of InAs geometry (A) and GaSb geometry (B) both grown on GaSb buffer. The values of d1 and d2 correspond to the crossing of E1 and H1 subbands, shown in Figs. 1B and 2B. (C and D) Fermi velocity vF of massless Dirac fermions as a function of GaSb layer thickness for the QWs of InAs geometry (C) and GaSb geometry (D) grown on different buffers: black curves, AlSb; red curves, GaSb; blue curves, InAs.

  • Fig. 6 Large-gap QSHI in three-layer QWs with InAs geometry.

    (A) Bandgap in the QSHI state for three-layer InAs/Ga0.6In0.4Sb QWs of InAs geometry as a function of d1 at d2 = 14 MLs (where one ML equals to a half of lattice constant in the bulk material). The black, red, and blue curves correspond to the structures grown on GaSb, AlSb, and Ga0.68In0.32Sb buffers, respectively. The inset shows the maximum gap, which can be achieved at a given value of d2, in the QW grown on Ga0.68In0.32Sb buffer. (B) Bandgap in the QSHI state for the InAs/Ga0.6In0.4Sb QWs with different values of d1 and d2 = 11 MLs as a function of strain in InAs layers ε1 and Ga0.6In0.4Sb layer ε2. Three specific cases for the values of ε1 and ε2, corresponding to the GaSb, AlSb, and Ga0.68In0.32Sb buffers, are marked by vertical dotted lines. (C and D) 3D plot of the band dispersion for the InAs/Ga0.6In0.4Sb QW grown on Ga0.68In0.32Sb buffer at d2 = 11 MLs, d1 = 27 MLs (C) and d1 = d1c (d1c ≈ 6 nm) (D). The x and y axes are oriented along (100) and (010) crystallographic directions, respectively.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/4/eaap7529/DC1

    Supplementary Text

    fig. S1. Comparison between calculations within the eight-band Kane model and by using the effective Hamiltonians for three-layer InAs/GaSb QWs.

    fig. S2. Bandgap in QSHI state in asymmetrical three-layer InAs/GaInSb QWs.

    table S1. Parameters involved in the effective Hamiltonian HInAs(kx, ky) for three-layer InAs/GaSb QWs with InAs geometry grown on GaSb buffer.

    table S2. Parameters of the effective 2D Hamiltonian HGaSb(kx, ky) for three-layer InAs/GaSb QWs with GaSb geometry grown on GaSb buffer.

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Text
    • fig. S1. Comparison between calculations within the eight-band Kane model and by using the effective Hamiltonians for three-layer InAs/GaSb QWs.
    • fig. S2. Bandgap in QSHI state in asymmetrical three-layer InAs/GaInSb QWs.
    • table S1. Parameters involved in the effective Hamiltonian HInAs(kx, ky) for three-layer InAs/GaSb QWs with InAs geometry grown on GaSb buffer.
    • table S2. Parameters of the effective 2D Hamiltonian HGaSb(kx, ky) for three-layer InAs/GaSb QWs with GaSb geometry grown on GaSb buffer.

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