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Ferroelastic modulation and the Bloch formalism

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Science Advances  07 Jun 2017:
Vol. 3, no. 6, e1602754
DOI: 10.1126/sciadv.1602754
  • Fig. 1 Structure of an orientation superlattice.

    Bottom: Periodic twinning of domain wedges A and B, each with crystalline symmetry Pmn21. Middle: Pmn21 symmetry unit cell. Top: OSL with symmetry P63cm.

  • Fig. 2 Free and localized modes of light.

    Perfectly localized and free modes for a ray launched parallel to the base or to the sides of wedges (a) and (b), respectively, that meet at the center of a hexagonal unit cell. Triangular wedge materials are chosen to be positively birefringent, with values of refractive indices ne ≅ 1.3938 and no = 1 for the extraordinary and ordinary refractions, respectively. The direction ⊥ uniaxis is depicted by the tick marks in each wedge. The wedge length l ≫ λ which is the wavelength of light for this case.

  • Fig. 3 Weak localization of light.

    Weakly localized mode for a ray launched at an angle of 1.749 × 10−2, with the base of the wedge in the hexagon shown at the top left. Inset shows ray propagation on a demagnified scale. The square in the inset is zoomed out in the main figure. The birefringent triangular wedges are identical to those in Fig. 2.

  • Fig. 4 Strong localization of light.

    Strongly localized mode for a ray launched at an angle of 6.981 × 10−4, with the side of the wedge in the hexagon shown at top center. Inset shows ray propagation on a demagnified scale. The square in the inset is zoomed out in the main figure. The birefringent triangular wedges are identical to those in Fig. 2.

  • Fig. 5 Bravais sublattices and orientational ordering.

    (A) The points in blue denote the Bravais lattice for the Pmn21-ordered wurtzite structure shown before orientational ordering. The tick marks depict the direction ⊥ uniaxis for wedges, and solid lines depict the unit cell. There will be similar lattices for the other two Pmn21-ordered variants. (B) Orientational ordering of the three ordered variants results in the original wurtzite Bravais lattice being differentiated into three different orientational Bravais sublattices (shown in red, blue, and green, respectively). The tick marks (which depict the direction ⊥ uniaxis for wedges) equivalently identify the three sublattices. The three interpenetrating flat tori for the P63cm symmetry OSL lattice are shown to the right of the OSL primitive unit cell.

  • Fig. 6 Energy ellipsoids and refraction.

    Shaded ellipses correspond to effective mass approximation free electron energy ellipses in the x,y plane of the P63cm symmetry OSL unit cell. The blue vector depicts an electron with a wave vector k in a wedge corresponding to a blue sublattice incident on an interface (blue-green line), with a wedge corresponding to a green sublattice. The green vector k′ depicts the refracted electron. Next, k′ incident on an interface (green-red line) with a wedge corresponding to a red sublattice gets refracted into the red vector k″. The wave vector component along the corresponding interface is preserved at each refraction.

  • Fig. 7 Spiral orbit for electrons.

    For out-of-plane momentum kz ≠0, the closed loop trajectory of Fig. 2 evolves into the spiral trajectory for electron propagation in a positive birefringence OSL, with γ = 1.3938.

  • Fig. 8 Poynting vector (or current density for Bloch states).

    (A) Bravais lattice for the OSL space group P63cm, with the outlined plane diagram formed from the hexagonal fundamental domain that was shown at the right of Fig. 5B, is composed of three interpenetrating flat tori. For l ≫ λde, a Bloch state ψk is identified in the blue rhombus with Pmn21 symmetry domains. Note that k changes can occur during interdomain propagation among the red, green, and blue sublattices. (Unlike in Fig. 5B, the sublattice Bravais points are not shown.) The Poynting vector (or current density Embedded Image) for ψk will exhibit refraction at the wedge interfaces. (B) Bravais lattice for the space group P63cm with the hexagonal unit cell outlined in black. For l ≯ λde, a Bloch state ΦK derived using the conventional Bloch formalism for the P63cm symmetry OSL is identified in the plane diagram. Because K is a constant of motion, the Poynting vector (or current density Embedded Image) for ΦK yields straight lines (no refraction).

  • Fig. 9 Tori representing tricolor and monochromatic Bravais lattices.

    (A) Torus for tricolor Bravais lattice using the plane diagram shown in Fig. 8A. Here, the translation group of Pmn21 is used for the reduction of this space group. (B) Torus for monochromatic Bravais lattice using the plane diagram in Fig. 8B. Here, the translation group of P63cm is used for the reduction of this space group

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/6/e1602754/DC1

    section S1. Light propagation in anisotropic media

    section S2. Electrons in a hexagonal OSL

    section S3. Photons in OSLs composed of positive birefringence wedges (Δn = neno > 0)

    section S4. The phase-matching condition for photons bound in a given loop

    section S5. Photons in OSLs composed of positive birefringence wedges

    section S6. Quantization condition for electron bound states

    section S7. Electrons in OSLs composed of positive uniaxial wedges

    section S8. Direct space and reciprocal lattice with unit cells

    fig. S1. Ray and wave normal directions.

    fig. S2. Consecutive uniaxial wedges A and B in the hexagonal unit cell of the OSL.

    fig. S3. Photons in OSLs composed of positive birefringence wedges (Δn = neno > 0).

    fig. S4. OSL with positive unaxial wedges.

    fig. S5. Direct space Bravais lattice and unit cells.

    fig. S6. Reciprocal space lattice and unit cells.

  • Supplementary Materials

    This PDF file includes:

    • section S1. Light propagation in anisotropic media
    • section S2. Electrons in a hexagonal OSL
    • section S3. Photons in OSLs composed of positive birefringence wedges (Δn = neno > 0)
    • section S4. The phase-matching condition for photons bound in a given loop
    • section S5. Photons in OSLs composed of positive birefringence wedges
    • section S6. Quantization condition for electron bound states
    • section S7. Electrons in OSLs composed of positive uniaxial wedges
    • section S8. Direct space and reciprocal lattice with unit cells
    • fig. S1. Ray and wave normal directions.
    • fig. S2. Consecutive uniaxial wedges A and B in the hexagonal unit cell of the OSL.
    • fig. S3. Photons in OSLs composed of positive birefringence wedges (Δn = neno > 0).
    • fig. S4. OSL with positive unaxial wedges.
    • fig. S5. Direct space Bravais lattice and unit cells.
    • fig. S6. Reciprocal space lattice and unit cells.

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