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Moiré excitons: From programmable quantum emitter arrays to spin-orbit–coupled artificial lattices

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Science Advances  10 Nov 2017:
Vol. 3, no. 11, e1701696
DOI: 10.1126/sciadv.1701696
  • Fig. 1 Moiré modulated local energy gaps and topographic height in the heterobilayer.

    (A) Long-period moiré pattern in an MoX2/WX2 heterobilayer. Green diamond is a supercell. Insets are close-ups of three locals, where atomic registries resemble lattice-matched bilayers of different R-type stacking. (B and C) Dependence of interlayer distance d on the atomic registries. In (C), dots are our first-principles calculations for the MoS2/WSe2 heterobilayer, and triangles are the scanning tunneling microscopy (STM) measured variation of the local d values in a b = 8.7 nm MoS2/WSe2 moiré in the study of Zhang et al. (26). The variation in d then leads to laterally modulated interlayer bias (∝ d) in a uniform perpendicular electric field, as (B) illustrates. (D) Schematic of relevant heterobilayer bands at the K valley, predominantly localized in either the MoX2 or WX2 layer. (E) Top: Variation of the local bandgap Eg [black arrow in (D)] in the MoS2/WSe2 moiré. Bottom: Variation of the local intralayer gaps [denoted by arrows of the same color in (D)]. In (C) and (E), the horizontal axis corresponds to the long diagonal of the moiré supercell, and the vertical axis plots the differences of the quantities from their minimal values. The curves are fitting of the data points using eqs. S2 and S3 in section S1.

  • Fig. 2 Nanopatterned spin optics of moiré excitons.

    (A) Left: Exciton wave packets at the locals with Embedded Image, Embedded Image, and Embedded Image registries, respectively (see Fig. 1A). Right: Corresponding Embedded Image transformation of electron Bloch function ψK,e, when the rotation center is fixed at a hexagon center in the hole layer. Gray dashed lines denote planes of constant phases in the envelope part of ψK,e, and red arrows denote the phase change by Embedded Image. (B) Left: Oscillator strength of the interlayer exciton. Right: Optical selection rule for the spin-up interlayer exciton (at the K valley). The distinct Embedded Image eigenvalues, as shown in (A), dictate the interlayer exciton emission to be circularly polarized at A and B with opposite helicity and forbidden at C. At other locals in the moiré, the emission is elliptically polarized (see inset, where ticks denote the major axis of polarization with length proportional to ellipticity). (C) Contrasted potential landscapes for the intra- and interlayer excitons, with the optical selection rule for the spin-up species shown at the energy minima. Transitions between the inter- and intralayer excitons (that is, via electron/hole hopping) can be induced by mid-infrared light with out-of-plane polarization.

  • Fig. 3 Electrically tunable and strain-tunable quantum emitter arrays.

    (A to C) Tuning of excitonic potential by perpendicular electric field (ε) in the R-type MoS2/WSe2 moiré. At zero field, nanodot confinements are at A points, realizing periodic array of excitonic quantum emitters, which are switched to B points at moderate field (see the main text). (D) Spin optical selection rule of quantum emitter at A. When loaded with two excitons, the cascaded emission generates a polarization-entangled photon pair. The optical selection rule is inverted when the quantum emitter is shifted to B [see (A) and (C)]. (E) Electric field tuning of exciton density of states (DOS) in the R-type MoS2/WSe2 moiré with b = 10 nm. The field dependence of V(A) and V(B) are denoted by the dotted blue and red lines on the field-energy plane. The colors of the two lowest energy peaks distinguish their different orbital compositions at A and B points in the moiré. (F) Exciton hopping integral between nearest-neighbor (NN) A and B dots in (B) (t0), between NN A dots (t1), and on-site exciton dipole-dipole (Udd) and exchange (Uex) interactions as functions of the moiré period b (see sections S4 and S5). The top horizontal axis is the corresponding lattice mismatch δ for rotationally aligned bilayer. (G) Exciton DOS at different b at zero electric field. The 20-meV scale bar applies for the energy axis in (E) and (G).

  • Fig. 4 Spin-orbit–coupled honeycomb lattices and Weyl nodes.

    (A) Opposite photon emission polarization at A and B sites and complex hopping matrix elements for the spin-up exciton. (B) Exciton spectrum at V(A) = V(B) and moiré period b = 10 nm, from the tight-binding model with the third NN hopping. t0 = 2.11 meV, t1 = 0.25 meV, and t2 = 0.14 meV. The bands feature a Dirac node and two Weyl nodes (highlighted by dotted circles). These magnetic monopoles are linked by an edge mode at a zigzag boundary, with spin polarization reversal at the Dirac node. Spin-down (spin-up) exciton is denoted by brown (blue) color. (C) Exact exciton spectrum in this superlattice potential (see section S4). Dirac/Weyl nodes are also seen in higher energy bands. (D) Schematic of the Dirac cones for the spin-up and spin-down excitons in the moiré–Brillouin zone (m-BZ), and edge modes at a zigzag boundary. Exciton-photon interconversion can directly happen within the shown light cone. (E) The Dirac and Weyl nodes are gapped by a finite A-B site energy difference Δ = 0.5t0, whereas the edge band dispersion is tuned by changing the on-site energy of the dots on the zigzag boundary [enclosed by the dashed box in (A)] by the amount U.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/11/e1701696/DC1

    section S1. Modulated electronic and topographic properties in the heterobilayer moiré

    section S2. Nanopatterned optical properties of the interlayer excitons in the moiré

    section S3. Complex hopping of the interlayer excitons in the moiré

    section S4. Exciton bands in superlattice potential: Exact solution and tight-binding model

    section S5. Exciton-exciton interactions in the superlattices

    fig. S1. Schematic of how the interlayer translation vector r0(R) (thick green arrows) changes as a function of in-plane position vector R.

    fig. S2. The modulations of layer separation δd, interlayer bandgap δEg, and intralayer bandgap δEintra for H-type MoS2/WSe2, R-type MoSe2/WSe2, and H-type MoSe2/WSe2 lattice-matched heterobilayers of various interlayer atomic registries.

    fig. S3. The potential profile of the interlayer excitons in the three types of TMD heterobilayers (see Eq. 1 in the main text).

    fig. S4. The ab initio results of the optical matrix elements at various interlayer translations r0.

    fig. S5. The real-space form of an interlayer exciton wave packet Formula, with width wb, corresponds to a Q-space distribution covering all the three main light cones (bright spots).

    fig. S6. Nanopatterned spin optics of moiré excitons in an H-type MoS2/WSe2 moiré pattern.

    fig. S7. The six reciprocal lattice vectors in the Fourier components of the excitonic potential, and the obtained hopping magnitudes t0,1,2 as functions of the moiré period b or V/ER.

    table S1. The parameters for fitting the first-principles results (symbols in fig. S2) with eqs. S2 and S3.

    table S2. The Formula quantum number of K-point Bloch function ψc or Formula for different rotation centers, taken from Liu et al. (29).

    table S3. The estimated radiative lifetimes for the interlayer exciton wave packets at A or B site in different heterobilayers with b = 15 nm.

    References (4856)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Modulated electronic and topographic properties in the heterobilayer moiré
    • section S2. Nanopatterned optical properties of the interlayer excitons in the moiré
    • section S3. Complex hopping of the interlayer excitons in the moiré
    • section S4. Exciton bands in superlattice potential: Exact solution and tightbinding model
    • section S5. Exciton-exciton interactions in the superlattices
    • fig. S1. Schematic of how the interlayer translation vector r0(R) (thick green arrows) changes as a function of in-plane position vector R.
    • fig. S2. The modulations of layer separation δd, interlayer bandgap δEg, and intralayer bandgap δEintra for H-type MoS2/WSe2, R-type MoSe2/WSe2, and H-type MoSe2/WSe2 lattice-matched heterobilayers of various interlayer atomic
      registries.
    • fig. S3. The potential profile of the interlayer excitons in the three types of TMD heterobilayers (compare Eq. 1 in the main text).
    • fig. S4. The ab initio results of the optical matrix elements at various interlayer translations r0.
    • fig. S5. The real-space form of an interlayer exciton wave packet X, with width w << b, corresponds to a Q-space distribution covering all the three main light cones (bright spots).
    • fig. S6. Nanopatterned spin optics of moiré excitons in an H-type MoS2/WSe2 moiré pattern.
    • fig. S7. The six reciprocal lattice vectors in the Fourier components of the excitonic potential, and the obtained hopping magnitudes t0,1,2 as functions of the moiré period b or V/ER.
    • table S1. The parameters for fitting the first-principles results (symbols in fig. S2) with eqs. S2 and S3.
    • table S2. The Cˆ3 quantum number of K-point Bloch function ψc or ψ*v for different rotation centers, taken from Liu et al. (29).
    • table S3. The estimated radiative lifetimes for the interlayer exciton wave packets at A or B site in different heterobilayers with b = 15 nm.
    • References (48–56)

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