Research ArticleCONDENSED MATTER PHYSICS

Multifunctional structural design of graphene thermoelectrics by Bayesian optimization

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Science Advances  15 Jun 2018:
Vol. 4, no. 6, eaar4192
DOI: 10.1126/sciadv.aar4192
  • Fig. 1 Nanostructured thermoelectric GNRs.

    (A) Periodically nanostructured GNR (model A) and (B) antidot GNR (model B). The size of the GNR section is determined by the index (Nx, Ny), where Nx and Ny are the numbers of carbon chains along the axial direction and the direction perpendicular to the axis, respectively. Binary flags represent the structural descriptor; 0 and 1, represent, respectively, defective and complete hexagonal lattices for model A and the pristine and antidot GNR sections for model B.

  • Fig. 2 Multifunctional structural optimization using model A.

    (A) Maximum (triangle), minimum (inverse triangle), and average (circle) ZT of all the candidates for m = 0 to 10. Dashed lines show the linear fitting in 1 ≤ m ≤ 10. (B) Efficiency of the Bayesian optimization. NBayes/rand is the average number of calculations needed until, for the first time, a structure that belongs to the top k% of all the candidates is found with the Bayesian/random search. (C and D) Optimal geometrical structure for P/Rth and its electron/phonon band structure with that of the pristine GNR. The P- and Rth-optimal structures are GNRs with m = 4 and 7, respectively. Bold lines in the electron and phonon band structures show the edge state and the longitudinal acoustic band, respectively. For the electron band structure, the Fermi level is set to zero.

  • Fig. 3 Optimization of the antidot GNR structure.

    (A) Thermoelectric properties—ZT, P, and Rth—of the representative structures: pristine (black), periodic (blue), and optimal (red) structures. The values are normalized by those of the pristine structure. (B) Periodic (top) and optimal aperiodic (bottom) structures. (C and D) Phonon/electron transmission functions, Θph/el(ω), of the representative structures. The Fermi level is set to zero for Θel(E). (E) Θel(E) (orange) and DOS in the nanostructured region (green) near the energy levels corresponding to the peak μ for ZT, μpeak, of the periodic (top) and optimal (bottom) structures; μpeak = −0.140 and −0.105 eV, respectively, as indicated by dashed lines. The band edge of the edge state is located at E = −0.078 eV corresponding to that of the Van Hove singularity, and black arrows and their labels indicate resonant states. (F) LDOS distribution of the resonant states of the periodic (top) and optimal (bottom) structures. The resonant numbers correspond to those in (E).

  • Fig. 4 Fluctuation of thermoelectric properties.

    P (blue), Rthph (gray), and Rth (red) of structures with the same numbers of antidots and homogeneous regions as the optimal structure. The thermoelectric properties are normalized by the value for the pristine GNR. The fitting curves of Gaussian distributions are shown as a reference.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/6/eaar4192/DC1

    section S1. Analysis method: Green’s function approach

    section S2. Efficiency of single-functional optimizations

    section S3. Acceleration of the Bayesian optimization

    section S4. Structural optimization of antidot armchair GNRs

    section S5. Phonon and electron transport in finite periodic structures

    section S6. Effects of resonant states on thermoelectric properties

    section S7. Resonant states in one-dimensional tight-binding chains

    section S8. Statistical errors of the efficiency of the Bayesian optimization

    section S9. Uniqueness of the optimal structures

    fig. S1. Comparison of optimization efficiency of different functionals.

    fig. S2. Acceleration of the Bayesian optimization using a topological descriptor: The mean shortest path.

    fig. S3. Structural optimization of antidot armchair GNR (AGNR).

    fig. S4. Phonon thermal resistance in finite periodic structures.

    fig. S5. Electron transport in finite periodic antidot structures.

    fig. S6. Effects of resonant states on thermoelectric properties.

    fig. S7. Electron transport properties of one-dimensional tight-binding chains.

    fig. S8. Efficiency of the Bayesian optimization.

    fig. S9. Model B structures that exhibit ZT > 0.95ZmaxT.

    References (7176)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Analysis method: Green’s function approach
    • section S2. Efficiency of single-functional optimizations
    • section S3. Acceleration of the Bayesian optimization
    • section S4. Structural optimization of antidot armchair GNRs
    • section S5. Phonon and electron transport in finite periodic structures
    • section S6. Effects of resonant states on thermoelectric properties
    • section S7. Resonant states in one-dimensional tight-binding chains
    • section S8. Statistical errors of the efficiency of the Bayesian optimization
    • section S9. Uniqueness of the optimal structures
    • fig. S1. Comparison of optimization efficiency of different functionals.
    • fig. S2. Acceleration of the Bayesian optimization using a topological descriptor: The mean shortest path.
    • fig. S3. Structural optimization of antidot armchair GNR (AGNR).
    • fig. S4. Phonon thermal resistance in finite periodic structures.
    • fig. S5. Electron transport in finite periodic antidot structures.
    • fig. S6. Effects of resonant states on thermoelectric properties.
    • fig. S7. Electron transport properties of one-dimensional tight-binding chains.
    • fig. S8. Efficiency of the Bayesian optimization.
    • fig. S9. Model B structures that exhibit ZT > 0.95ZmaxT.
    • References (71–76)

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