Methods
Kohm-Sham DFT simulations were carried out using the Gaussian03TM package,
Revision C.02 [M1]. In order to optimize the structures used in the present work, we used
the PBE-PBE generalized gradient approximation for the exchange-correlation term
[M2], and Troullier-Martin potentials [M3]. Optimizations have been carried out using
the 3-21G basis set [M4] and an 8 x 8 x 8 Monkhorst-Pack mesh grid, equally spaced in
the k-space [M5]. The use of the 3-21G basis set is motivated by the complexity of the
calculations to be performed, requiring the optimization of several transition structures at
multiple pressures. Single-point energy calculations using more sophisticated basis sets,
6-31G(d, p) and 6-311G(d, p), were also performed and shown to never affect the
stability of any polytype with respect to any other. In particular, at room conditions, the
difference in SCF energy between B11Cp(CBC) and B12(CCC) polytypes depends on the
chosen basis set for less than 5%.
Although the Gaussian package is not tailored for use in highly metallic systems,
it has been proven to correctly deal with weakly metallic systems such as graphite (for
instance in metallic carbon nanotubes [M6], where the obtained results [M7] have been
found to be consistent with those from codes utilizing more sophisticated smearing
utilities). Furthermore, in the present work, we have been able to exploit some improved
capabilities associated with the more recent Gaussian03 version. Specifically, a Fermi
broadening fractional occupation number method [M8] was used in all our simulations in
order to deal with electronic bands touching or approaching the Fermi level.
The Gibbs free energies of the various polytypes were provided by the
thermochemistry utilities [M9] of the Gaussian package. In brief, from the total energy
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per atom (E0) we calculated the Helmholtz free energy per site F ≈ E0 + U0 + Fvib(T)
where U0 is the zero-point energy, and Fvib the temperature-dependent vibrational energy
within the quasi-harmonic approximation. For both, we have used a Debye
approximation of the vibrational density-of states. From that, the relevant quantity to test
a polytype stability, the Gibbs free energy G = F + P⋅V, can be easily extracted for all
polytypes, where V is the unit cell volume divided by the number of atoms in the unit
cell. The as calculated free energies have been already inclusive of the different levels of
degeneracy of the various polytypes. It must be noted that the choice of using the Gibbs
free energy per site in estimating the phase population probabilities holds since the
disorder affects the material at an atomistic level during its out-of-equilibrium formation.
In contrast, the coexistence of the various phases at equilibrium conditions is controlled
by disorder at a crystallographic level, hence, it requires the use of the Gibbs energy per
unit cell as a reference quantity.
Room pressure elastic constants of B4C were determined theoretically for the first
time by Lee et al [M10]. Ultrasonic experiments by McClellan et al [M11] gave data in
good agreement with the theory (uncertainty: ±20%). By calculating the force on the
atoms and their derivatives, we obtained force constants within such an uncertainty range
for B11Cp(CBC), B11Ce(CBC), B11Ce(BCC) and B12(CCC) at room-pressure (10-4 GPa).
Through the elastic constants, we have then determined the changes in lattice parameters
occurring at 40 GPa stress and subsequently we used such lattice parameters for new
structure optimization. The as obtained free energies agree well with predictions by the
Birch-Murnaghan high-pressure equation of state [M12]. We have then used such an
equation to interpolate the Gibbs free energy between 10-4 and 40 GPa. The transition
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from the boron carbide polytypes and the B12/Graphite phase have been determined in
two steps. First, atomic coordinates corresponding to local minima along the
corresponding reaction path have been determined, through stability calculations, to
correspond to the B11Ce(CBC), B11Ce(CCB) and B12(CCC) polytypes. Second, a QST3
calculation [M13] allowed us to optimize the intermediate first order saddle point and
calculate the correspondent transition energies shown in Fig. 3.
3
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