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We present an
*ab-initio*, self-consistent density functional theory (DFT) description of ground state electronic and related properties of hexagonal boron nitride (h-BN). We used a local density approximation (LDA) potential and the linear combination of atomic orbitals (LCAO) formalism. We rigorously implemented the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). The method ensures a generalized minimization of the energy that is far beyond what can be obtained with self-consistency iterations using a single basis set. The method leads to the ground state of the material, in a verifiable manner, without employing over-complete basis sets. We report the ground state band structure, band gap, total and partial densities of states, and electron and hole effective masses of hexagonal boron nitride (h-BN). Our calculated, indirect band gap of 4.37 eV, obtained with room temperature experimental lattice constants of
*a* = 2.504
Å and
*c *= 6.661
Å, is in agreement with the measured value of 4.3 eV. The valence band maximum is slightly to the left of the K point, while the conduction band minimum is at the M point. Our calculated, total width of the valence and total and partial densities of states are in agreement with corresponding, experimental findings.

The demand for compact ultraviolet laser devices has led many researchers to search for materials with band gaps larger than that of GaN (3.4 eV), a material presently utilized in the fabrication of high-power, blue-ray laser devices [

As shown in

Experimental method | E_{g} (eV) |
---|---|

X-ray photoemission spectra | 3.6 [a], 3.85 [b] |

Optical and UV absorption | 3.9 [c], 4.3 [d] |

Laser-induced fluorescence (LIF) | 4.02 [e] |

Optical reflectivity spectra | 4.5 [f], 5.2 [g] |

Luminescence optical spectra | >5.5 [h], 5.89 [i], 5.95 [j] |

Photoconductivity, and absorption spectra | 5.8 [k], 5.83 [l] |

Temperature dependence of electrical resistivity | 7.1 [m] |

^{a}Ref. [^{b}Ref. [^{c}Ref. [^{d}Ref. [^{e}Ref. [^{f}Ref. [^{g}Ref. [^{h}Ref. [^{i}Ref. [^{j}Ref. [^{k}Ref. [^{l}Ref. [^{m}Ref. [

Computational method | Potentials | E_{g} (eV) |
---|---|---|

Linear Combination of Pseudoatomic Orbitals (LCPAO) | LDA | 3.7 (M-H) [a] |

FP-LAPW | LDA | 3.9 (H-M) [b] 4.3 (H-H) [b] |

Ab-initio Pseudopotential | LDA | 3.9 (K-M) [c] |

OLCAO | LDA | 4.07 (M-K) [d] |

Ultra soft Pseudopotential | LDA | 4.1 (H-M) [e] 4.5 (M-M) [e] |

FP-LAPW | LDA | 4.0 (H-M) [f] 4.5 (M-M) [f] |

FP-LAPW | LDA | 4.58 (H-K) [g] |

FP-LAPW | PW91-GGA | 4.53 (Γ-K) [g] |

FP-LAPW | PBE-GGA | 4.54 (Γ-K) [g] |

Projected-Augmented-Wave (PAW) | LDA | 4.02 (K-M) [h] |

PAW (VASP) | LDA | 4.21 (H-M) [i] |

PAW (VASP) | GGA | 4.39 (H-M) [i] |

PAW | GGA | 4.47 (K-M) [j] |

GW | GGA | 5.4 (H-M) [c] |

GW | LDA | 5.95 (K-M) [h] |

GW | LDA | 5.95 (H-M) [k] |

^{a}Ref. [^{b}Ref. [^{c}Ref. [^{d}Ref. [^{e}Ref. [^{f}Ref. [^{g}Ref. [^{h}Ref. [^{i}Ref. [^{j}Ref. [^{k}Ref. [

linear combination of pseudo-atomic-orbitals (PAO) method to calculate properties of h-BN. Their calculated, indirect band gap, from H to M, was 3.7 eV [

Clearly, this range of theoretical results for the band gap of h-BN, including the seven (7) different pairs of VBM-CBM, points to the need for further work. Additionally, and unlike the cases for most semiconductors, the experimental results in

We succinctly provide below the essential features of our computational approach. Extensive details on it are available in the literature [

As per the second DFT theorem, self-consistent iterations with a single basis set lead to a stationary solution among an infinite number of such solutions. This fact resides in the reality that the ground state charge density (i.e. basis set) is not à priori known, as far as we can determine. Consequently, the chances are extremely small for a calculation with a single basis set to lead to the ground state of the system or to avoid over-complete basis sets.

We have described in previous publications a straightforward way to search for and to reach the ground state of the system. Beginning with a small basis set that is large enough to account for all the electrons in the system, we perform successive self-consistent calculations, where the basis set of a calculation, except for the first one, is that of the preceding calculation augmented with one orbital. The first and second versions of our method, known as BZW and BZW-EF method, differ as follows. For the first one, we add orbitals in the order of increasing energy of the excited states they represent. In the second, we heed the “arbitrary variations” clause of the second DFT theorem and add orbitals so as to recognize the primacy of polarization orbitals (p, d, and f) over the spherical symmetry of s orbitals for valence electrons. Indeed, for diatomic and any other multi-atomic system, valence electrons do not possess any full, spherical symmetry known to us, unlike the core electrons. The above referenced, successive calculations continue until three (3) consecutive ones produce the same occupied energies. This criterion guarantees the attainment of the absolute minima of the occupied energies (i.e. the true ground state). With just two (2) consecutive calculations leading to the same occupied energies, these energies could represent a local minima and not the absolute ones. The first of the referenced three (3) consecutive calculations [

In this study, we utilized the program package developed at the US Department of Energy’s Ames Laboratory, in Ames, Iowa. B and N are light enough to neglect relativistic corrections. Self-consistent calculations of the electronic energies and wave functions for the atomic or ionic species provided input data for the solid-state calculations. Specifically, for hexagonal BN, the species we considered were B^{3+} and N^{3−}. Preliminary calculations for neutral atoms (B and N) pointed to a charge transfer larger than 2, from B to N.

We provide below computational details to enable the replication of our work. Hexagonal BN (h-BN) belongs to the D 6 h 4 space group, with a space group number of 194, a Pearson symbol of hP4, and Patterson space group P6_{3}/mmc [_{B}, where a_{B} is the Bohr radius) and c = 6.661 Å = 12.5875 a.u. at room temperature. We expanded the radial parts of the orbitals in terms of even-tempered Gaussian functions. The s and p orbitals for the cation B^{3+} were each described with 16 even-tempered Gaussian functions with the respective minimum and maximum exponents of 0.2658 and 1.655 × 10^{4} for the atomic potential and 0.1242 and 1.365 × 10^{4} for the atomic wave functions. The self-consistent calculations for B^{3+} led to the total charge of 2.0005, which is also the valence charge, with an error per electron of 2.5 × 10^{−4}. Similarly, the s and p orbitals for N^{3−} were described with 20 even-tempered Gaussian functions with the respective minimum and maximum exponents of 0.1600 and 1.600 × 10^{4} for the atomic potential and 0.1000 and 1.300 × 10^{4} for the atomic wave functions. These exponents led to the convergence of the atomic calculations for N^{3−} with the total, core and valence charges of 10.00004, 2.00002, and 8.00002, respectively. The error per electron was therefore 4 × 10^{−6}. We utilized a 24 k-point mesh with proper weights, in the irreducible Brillouin zone, for the self-consistency iterations. The criterion for the convergence of the iterations was a difference of 10^{−5} or less between the potentials from two consecutive ones. We used 140 k points in the irreducible Brillouin zone for the production of the final, self-consistent bands.

Calculation No. | Valence Orbitals for B^{3+} | Valence Orbitals for N^{3−} | No. of Functions | Band Gaps (eV) (near K-M) |
---|---|---|---|---|

I | 1s^{2}2p^{0}2s^{0} | 2s^{2}2p^{6} | 36 | 7.499 |

II | 1s^{2}2p^{0}2s^{0} | 2s^{2}2p^{6}3p^{0} | 48 | 5.767 |

III | 1s^{2}2p^{0}2s^{0}3p^{0} | 2s^{2}2p^{6}3p^{0} | 60 | 4.370 |

IV | 1s^{2}2p^{0}2s^{0}3p^{0} | 2s^{2}2^{6}3p^{0}3s^{0} | 64 | 4.369 |

V | 1s^{2}2p^{0}2s^{0}3p^{0}3s^{0} | 2s^{2}2^{6}3p^{0}3s^{0} | 68 | 4.365 |

VI | 1s^{2}2p^{0}2s^{0}3p^{0}3s^{0} | 2s^{2}2p^{6}3p^{0}3s^{0}4p^{0} | 80 | 4.210 |

material. The basis set for this calculation is the optimal basis set, i.e. the smallest basis set leading to the ground state of the material, without being over-complete.

Figures 1(a)-(e) provide a graphical illustration of the generalized minimization of the energy, as the basis set is methodically augmented for successive, self-consistent calculations. Every pair of bands from consecutive calculations is shown below. In

The top of the valence band (VBM) is between K and Γ, at equally 10% of the K-Γ separation, to the left of K. Its distance from K is ∆K = (4π/3a) × 0.1 = 0.0885, where a = 4.7319 a.u. is a lattice constant in atomic units. Hence, the location of the VBM is at K* = K − ΔK = (0, 0.7965, 0), to the left of K.

Even though the occupied energies in

Γ-point | K-ΔK-point | K-point | H-point | A-point | M-point | L-point |
---|---|---|---|---|---|---|

17.357 | 21.116 | 20.759 | 19.791 | 16.4508 | 21.593 | 21.745 |

16.617 | 20.969 | 20.759 | 19.791 | 16.4508 | 21.259 | 21.745 |

13.322 | 19.793 | 18.939 | 18.802 | 13.320 | 21.033 | 18.820 |

13.321 | 18.613 | 18.939 | 18.802 | 13.320 | 20.236 | 18.820 |

13.305 | 16.896 | 17.668 | 14.843 | 13.319 | 15.699 | 13.856 |

13.304 | 14.445 | 13.994 | 14.843 | 13.319 | 12.780 | 13.856 |

13.056 | 13.656 | 13.994 | 13.878 | 12.958 | 10.689 | 10.824 |

12.592 | 12.309 | 12.957 | 13.878 | 12.958 | 10.163 | 10.824 |

9.714 | 5.445 | 5.064 | 4.715 | 7.263 | 6.222 | 5.040 |

5.049 | 4.953 | 5.064 | 4.715 | 7.263 | 4.369 | 5.040 |

−2.419 | 0.000 | −0.138 | −0.048 | −2.435 | −0.482 | −1.007 |

−2.420 | −0.614 | −0.138 | −0.048 | −2.435 | −1.552 | −1.007 |

−2.453 | −7.827 | −8.067 | −8.082 | −2.436 | −5.399 | −5.423 |

−2.453 | −7.833 | −8.067 | −8.082 | −2.436 | −5.452 | −5.423 |

−4.365 | −9.241 | −9.242 | −9.322 | −5.606 | −9.960 | −9.990 |

−6.593 | −9.366 | −9.400 | −9.322 | −5.606 | −10.012 | −9.990 |

−18.206 | −14.748 | −14.653 | −14.653 | −18.368 | −15.254 | −15.283 |

−18.509 | −14.801 | −14.653 | −14.653 | −18.368 | −15.313 | −15.283 |

subtilities relative to the valence band maximum (VBM) and the conduction band minimum (CBM). In particular, our close examination of the bands hints at a possible explanation of the multitude of VBM-CBM pairs reported by previous density functional theory calculations. These calculations, as far as we can determine, did not perform the generalized minimization of the energy as dictated by the second DFT theorem.

The lower and upper groups of valence bands have widths of 3.98 eV and 10.02 eV, respectively. Three major peaks in the density of states for the conduction bands are located at 4.92 eV, 12.88 eV, and 18.46 eV. The above characteristics of the total density of states (DOS), for h-BN, will be hopefully confirmed by future experimental measurements. Additionally, the eigenvalues in

Several transport properties, including various mobilities for electrons or holes, depend on the inverse of the electron or hole effective masses, respectively. For this reason, we have calculated the electron and hole effective masses shown in _{0}. With values of 0.205m_{0}, 2.250m_{0}, and 1.730m_{0} in the M to Γ, M to K, and M to L directions, respectively, the electron effective mass at the bottom of the conduction band is clearly anisotropic. The same is true for the electron effective mass at H, even though its values from H to A and H to Γ are identical.

The hole effective masses from K* to Γ, K* to H, and K* to M are respectively 0.534, 0.569, and 1.48, in units of m_{0}. The calculated hole effective masses at the H symmetry point, along H-A, H-Γ, H-K, and H-L axes, are 0.822, 0.822, 3.468, and 1.671, respectively, in units of m_{0}. These hole effective masses are anisotropic, despite the equality of the ones from H to A and H to Γ.

A discussion of our results, particularly in relation to findings from previous DFT calculations, rests on the following fact. None of the previous calculations

Types and Directions of Effective Masses | Values of Effective Masses (m_{0}) |
---|---|

M_{e} (M-Γ) | 0.205 |

M_{e} (M-K) | 2.250 |

M_{e} (M-L) | 1.730 |

M_{e} (H-A) | 0.588 |

M_{e} (H-Γ) | 0.588 |

M_{e} (H-K) | 1.102 |

M_{e} (H-L) | 3.129 |

M_{e} (K-Γ) | 0.387 |

M_{e} (K-H) | 0.433 |

M_{h} (K*-Γ) | 0.534 |

M_{h} (K*-H) | 0.569 |

M_{h} (K*-M) | 1.480 |

M_{h} (H-A) | 0.822 |

M_{h} (H-Γ) | 0.822 |

M_{h} (H-K) | 3.468 |

M_{h} (H-L) | 1.671 |

appear to have performed a generalized minimization of the energy. The minimization obtained following self-consistent iterations, with a single basis set, produces the minimum of the energy relative to that basis set. Such solutions are stationary ones whose number is practically infinite. None should be à priori assumed to provide a description of the ground state of the material. Consequently, the computational results should not be expected to possess the full, physical content of DFT or to agree with experimental measurement. Our generalized minimization, as thoroughly explained above, verifiably leads to the absolute minima of the occupied energies, i.e. the ground state, as required by the second DFT theorem. Explicitly searching for the ground state and avoiding basis sets that are overcomplete for the description of the ground state are two requirements for a correctly performed DFT calculation. We address below plausible, negative consequences use of over-complete basis sets.

With the second corollary of the first DFT theorem, i.e. that the spectrum of the Hamiltonian is a unique functional of the ground state charge density [

With the above understanding, we discuss the fine structures of the bands using the enlarged graphs in

is slightly larger than the one at H which is 4.763 eV + 0.048 eV = 4.811 eV.

The above fine structures of the bands hint to a possible explanation of the report of seven (7) different VBM-CBM pairs by previous DFT calculations. Indeed, while the presumed single basis sets in these calculations may be close to or contain the corresponding optimal basis sets, with the above subtle features of the band structure, the slightest deviation of these basis sets from the one describing the ground state could explain the differences between the resulting bands and between them and the ones reported here. Additionally, without the generalized minimization, it is practically hopeless to have the basis set complete for the description of the ground state, without being over-complete.

We have presented the description of electronic and related properties of the ground state of h-BN, as obtained from ab-initio, self-consistent density functional theory (DFT) calculations. Our generalized minimization of the energy, following the BZW-EF method, verifiably led to the ground state and avoided over-complete basis sets. Our findings possess the full, physical content of DFT. Our calculated indirect band gap from K* to M is 4.37 eV. This value is practically in agreement with the experimental finding of 4.30 eV which is the most accepted one in the literature. The density of states (DOS) and partial densities of states (p-DOS) are in good agreement with those from electron momentum spectroscopy (EMS) [

This work was funded in part by the National Science Foundation [NSF, Award Nos. EPS-1003897, NSF (2010-2015)-RH SUBR, and HRD-1002541], the US Department of Energy, National Nuclear Security Administration (NNSA, Award No. DE-NA0002630), LaSPACE and LONI-SUBR.

The authors declare no conflicts of interest regarding the publication of this paper.

Malozovsky, Y., Bamba, C., Stewart, A., Franklin, L. and Bagayoko, D. (2020) Accurate Ground State Electronic and Related Properties of Hexagonal Boron Nitride (h-BN). Journal of Modern Physics, 11, 928-943. https://doi.org/10.4236/jmp.2020.116057