Research ArticleChemistry

Polymorphism of bulk boron nitride

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Science Advances  18 Jan 2019:
Vol. 5, no. 1, eaau5832
DOI: 10.1126/sciadv.aau5832
  • Fig. 1 BN polymorphs and corresponding zero-temperature energies calculated with first-principles–based methods.

    The energy of the h-BN(AA′) polymorph is taken as the reference value in each of the series. “D3” (13, 14), “FI” (17), “D3(BJ)” (15), “RPA” (2226), and “MBD” (16) stand for different dispersion-corrected first-principles methods, all taken over PBE (10) as a base functional. LDA (9) and PBE methods provide results outside the selected range (table S2). Energy results include quantum nuclear effects through zero-point energy (ZPE) corrections as calculated with the LDA method (Methods). The notation used to refer to the BN polymorphs throughout the text along with the corresponding space groups and crystal structures are specified; the letters within parentheses accompanying the hexagonal polymorphs indicate the stacking sequence between consecutive B–N planes along the hexagonal c axis.

  • Fig. 2 Monoclinic phase m-BN (space group Cm) reported in this study.

    (A to D) Projections showing its similarities to the h-BN polymorph. (E) Key structural parameters found via different methods (table S3). Boron and nitrogen atoms are represented with green and blue spheres, respectively.

  • Fig. 3 Gibbs free energy of BN polymorphs at zero pressure expressed as a function of temperature.

    A phase transition between the c-BN and h-BN(AA′) polymorphs is predicted to occur at Tc→h = 335 ± 30 K. Temperature-induced volume expansion effects are appropriately taken into account (Methods and fig. S2). The mass densities of the two polymorphs at the transition temperature are indicated along with the corresponding experimental room temperature values [within parentheses, taken from previous works (30, 31)]. The shaded area indicates the numerical error of ±0.26 kJ/mol in the RPA and vibrational free energy calculations, which leads to the ±30-K error in Tc→h, indicated by the horizontal bar.

  • Table 1 Elastic constants and bulk modulus in the Voigt-Reuss-Hill (VRH) approximation [as this is appropriate for polycrystalline samples (36)] of the most stable polymorphs calculated with the LDA (9) and RPA (2226) methods.

    LDA results are very close to those from RPA, showing the suitability of LDA for estimating second energy derivatives. Results are in units of gigapascal.

    C11C22C33C12C23C13BVRH
    c-BNLDA997997997101101101402
    c-BNRPA968968968818181378
    h-BN(AA′)LDA9239262817433138
    h-BN(AA′)RPA91091529153−4−5131
    m-BNLDA7518394515610068160
    m-BNRPA745830431379058151

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/1/eaau5832/DC1

    Supplementary Methods

    Fig. S1. Phonon spectrum of the new monoclinic phase m-BN (space group Cm) reported in this study as calculated with the LDA method at zero pressure.

    Fig. S2. Gibbs free energy differences among several BN polymorphs calculated at zero pressure and expressed as a function of temperature.

    Fig. S3. Phonon spectrum of the c-BN (space group Formula) and h-BN(AA′) (space group P63/mmc) polymorphs at zero pressure as calculated with the LDA method.

    Fig. S4. Convergence tests of the electronic and vibrational free energies calculated with standard DFT methods (i.e., PBE-D3) in the h-BN(AA′) polymorph.

    Fig. S5. ZPE corrections accounting for zero-temperature quantum nuclear effects (27) in several BN polymorphs as calculated with the LDA (9) and PBE-D3 (13, 14) methods at zero pressure.

    Table S1. Numerical tests performed for stage 3 in the RPA calculations.

    Table S2. Zero-temperature electronic energies (i.e., neglecting zero-point energy corrections) of several BN polymorphs as compared to that of the c-BN (space group Formula) phase.

    Table S3. Structural properties of the monoclinic phase m-BN (space group Cm) reported in this study as calculated with different methods based on DFT.

    Table S4. Elastic constants associated with compressive deformations Cij’s, bulk modulus, BVRH, shear modulus, GVRH, isotropic Poisson’s ratio, μVRH, and Young’s modulus, YVRH, calculated in the Voigt-Reuss-Hill approximation [“VRH,” as this is appropriate for polycrystalline samples; see previous work (30) for the corresponding analytical expressions] with the LDA method.

    References (37, 38)

  • Supplementary Materials

    This PDF file includes:

    • Supplementary Methods
    • Fig. S1. Phonon spectrum of the new monoclinic phase m-BN (space group Cm) reported in this study as calculated with the LDA method at zero pressure.
    • Fig. S2. Gibbs free energy differences among several BN polymorphs calculated at zero pressure and expressed as a function of temperature.
    • Fig. S3. Phonon spectrum of the c-BN (space group F4¯3m) and h-BN(AA′) (space group P63/mmc) polymorphs at zero pressure as calculated with the LDA method.
    • Fig. S4. Convergence tests of the electronic and vibrational free energies calculated with standard DFT methods (i.e., PBE-D3) in the h-BN(AA′) polymorph.
    • Fig. S5. ZPE corrections accounting for zero-temperature quantum nuclear effects (27) in several BN polymorphs as calculated with the LDA (9) and PBE-D3 (13, 14) methods at zero pressure.
    • Table S1. Numerical tests performed for stage 3 in the RPA calculations.
    • Table S2. Zero-temperature electronic energies (i.e., neglecting zero-point energy corrections) of several BN polymorphs as compared to that of the c-BN (space group F4¯3m) phase.
    • Table S3. Structural properties of the monoclinic phase m-BN (space group Cm) reported in this study as calculated with different methods based on DFT.
    • Table S4. Elastic constants associated with compressive deformations Cij’s, bulk modulus, BVRH, shear modulus, GVRH, isotropic Poisson’s ratio, μVRH, and Young’s modulus, YVRH, calculated in the Voigt-Reuss-Hill approximation “VRH,” as this is appropriate for polycrystalline samples; see previous work (30) for the corresponding analytical expressions with the LDA method.
    • References (37, 38)

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