Research ArticleCONDENSED MATTER PHYSICS

Determination of layer-dependent exciton binding energies in few-layer black phosphorus

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Science Advances  16 Mar 2018:
Vol. 4, no. 3, eaap9977
DOI: 10.1126/sciadv.aap9977
  • Fig. 1 Optical absorption in 2D semiconductors.

    (A) Lattice structure of few-layer BP, showing the puckered hexagonal crystal with two characteristic directions: AC and ZZ. (B) Illustration of optical absorption in a model conventional 2D semiconductor, including optical transitions to excitonic ground (1s) and excited states (2s), as well as continuum states above the quasi-particle bandgap Eg. The exciton resonance is characteristic of a Lorentzian line shape, whereas the continuum absorption exhibits a step-like feature. (C) IR extinction spectra (1 − T/T0) for a representative 4L BP sample on a PDMS substrate, with two different light polarizations. The symbol E11 denotes exciton relating to optical transition v1→c1, as illustrated in Fig. 2H. Data were collected at room temperature.

  • Fig. 2 Layer-dependent IR extinction spectra.

    (A to G) IR extinction spectra (1 − T/T0) for few-layer BP on PDMS substrates with layer number N = 2 to 8, with incident light polarized along the AC direction. The red arrows indicate the lowest-energy excited states (2s) of the E11 transition. For 7L and 8L samples, the 2s peaks are unresolvable. (H) Schematic illustration of optical transitions between quantized subbands in few-layer BP, with the symbols E11 and E22 denoting transitions v1→c1 and v2→c2, respectively.

  • Fig. 3 Extraction of the exciton binding energies and quasi-particle bandgaps.

    (A) Experimentally and theoretically obtained 1s-2s separation (Δ12) as a function of layer number. The blue diamonds denote the exciton binding energy obtained using a numerical method detailed in the Supplementary Materials. (B) Layer dependence of the 1s and 2s transition energies and the quasi-particle bandgap. The 1s and 2s transition energies are obtained experimentally, as indicated in Fig. 2. The quasi-particle bandgap is deduced as Eg = Eopt + Eb, where Eopt is the 1s transition energy.

  • Fig. 4 Scaling behavior of the exciton binding energy with layer number.

    The black diamonds are theoretical values for few-layer BP with N = 1 to 8 obtained within the Wannier-Mott framework. The red line is a fit to the data, using the formula Eb = 3/4παeff. αeff is the effective 2D polarizability, with a linear function of layer number N: αeff = α0 + Nα1, where α0 and Nα1 describe the screening effect from the underlying substrate and the 2D material itself, respectively.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/3/eaap9977/DC1

    note S1. Anisotropic optical absorption

    note S2. Numerical calculation of the exciton binding energy

    note S3. Substrate screening effect on the exciton binding energy

    fig. S1. Additional IR extinction data for 4L and 6L BP.

    fig. S2. Polarized IR extinction spectra for an 8L BP.

    fig. S3. Comparison of exciton binding energies for suspended and supported few-layer BP.

    table S1. Summary of experimentally and theoretically obtained values for 1s-2s separation (Δ12), as well as the exciton binding energy (Eb).

    References (3943)

  • Supplementary Materials

    This PDF file includes:

    • note S1. Anisotropic optical absorption
    • note S2. Numerical calculation of the exciton binding energy
    • note S3. Substrate screening effect on the exciton binding energy
    • fig. S1. Additional IR extinction data for 4L and 6L BP.
    • fig. S2. Polarized IR extinction spectra for an 8L BP.
    • fig. S3. Comparison of exciton binding energies for suspended and supported few-layer BP.
    • table S1. Summary of experimentally and theoretically obtained values for 1s-2s separation (Δ12), as well as the exciton binding energy (Eb).
    • References (39–43)

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