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We report details of our
*ab-initio*, self-consistent density functional theory (DFT) calculations of electronic and related properties of wurtzite beryllium oxide (w-BeO). Our calculations were performed using a local density approximation (LDA) potential and the linear combination of atomic orbitals (LCAO) formalism. Unlike previous DFT studies of BeO, the implementation of the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by the work of Ekuma and Franklin (BZW-EF), ensures the full physical content of the results of our calculations, as per the derivation of DFT. We present our computed band gap, total and partial densities of states, and effective masses. Our direct band gap of 10.30 eV, reached by using the experimental lattice constants of a = 2.6979
Å and c = 4.3772
Å at room temperature, agrees very well the experimental values of 10.28 eV and 10.3 eV. The hybridization of O and Be p states in the upper valence bands, as per our calculated, partial densities of states, are in agreement with corresponding, experimental findings.

Beryllium oxide, also known as beryllia, has attracted much interest due to its current and potential applications. Second only to diamond among electrically insulating materials, BeO is an excellent thermal conductor with a measured room temperature band gap of as high as an estimated 10.63 eV [

Several sources have reported results of experimental studies of w-BeO as indicated in

While the band gap is somewhat established experimentally, previous theoretical calculations reported band gaps over a wide range. The calculated values disagree among themselves and with experiment as can be seen in

Experimental works | Eg (eV) |
---|---|

Optical absorption edge on single crystal BeO at 300 K | 10.3^{3} |

Normal-incidence reflectance of wurtzite BeO at room temperature (estimated band gap value) | 10.63 ± 0.10^{1 } |

Far-ultraviolet reflectance at room temperature | 10.28^{5} and 10.30^{2} |

Electron energy loss measurements for amorphous and polycrystalline BeO thin films | 9.6^{5} |

Computational method | Potentials (DFT and Other) | Eg (eV) |
---|---|---|

Non local pseudopotential | LDA | 7.0^{a} |

LMTO-ASA and PAW | LDA | 7.23^{b} |

Nonlocal, norm-conserving pseudopotential | LDA | 7.36^{c} |

Plane wave pseudopotential | LDA | 7.36^{d} |

SIC-pseudopotential | LDA | 7.41^{e} |

Plane wave pseudopotential | LDA | 7.42^{f} |

OLCAO | LDA | 7.54^{g} |

Nonlocal empirical pseudopotential | LDA | 7.63^{h} |

Ab initio pseudopotential | LDA | 7.8^{i} |

Plane wave basis method | GGA | 6.99^{j} |

FP-LAPW + lo | GGA | 7.44^{k} |

FP-LAPW + lo | GGA | 7.47^{l} |

Pseudopotential-PAW | GGA | 7.48^{m} |

Plane wave pseudopotential | GGA | 7.50^{n} |

Plane wave pseudopotential | GGA | 7.66^{o} |

Nonlocal empirical pseudopotential | GGA | 7.70^{h} |

PBE | GGA | 8.4^{p} |

FP + LAPW + lo | GGA | 8.4^{q} |

FP-LAPW + lo | EV-GGA | 8.57^{k} |

LMTO-ASA | GGA | 9.50^{r} |

B3LYP | GGA | 10.39^{s} |

^{a}Reference [^{b}Reference [^{c}Reference [^{d}Reference [^{e}Reference [^{f}Reference [^{g}Reference [^{h}Reference [^{i}Reference [^{j}Reference [^{k}Reference [^{m}Reference [^{n}Reference [^{o}Reference [^{p}Reference [^{q}Reference [^{r}Reference [^{s}Reference [

ab-initio LDA potential [

Given the above theoretical results, listed in

More details about our computational approach are available in previous articles [

We used a computer program originated in the US Department of Energy’s Ames Laboratory, in Iowa, to perform non-relativistic calculations. We first performed the LCAO self-consistent calculations of the electronic energy at the atomic or ionic level of the material being investigated. These species are Be^{2+} and O^{2−}. Our choice was based on the results of preliminary calculations for w-BeO using neutral atoms. The preliminary results showed a transfer of approximately two electrons from Be to O. We subsequently performed ab-initio; self-consistent calculations to determine the atomic basis set to be used in the solid-state calculations for Be^{2+} and O^{2−}.

As per by the BZW-EF method, our self-consistent calculations began with a small basis set that contained the minimum basis set (MBS); the MBS is the one just big enough to encompass all the electrons in the system. The basis set of the second calculation is composed of the basis set from Calculation I augmented with an appropriate orbital that represents an excited energy level in the ionic species of the system. Then, we compared numerically and graphically the occupied energies from the first (I) and second (II) calculations, after setting the Fermi level to zero. The comparison showed that some valence bands from Calculation II were lower than their corresponding ones from Calculation I. This is the indication that the basis set from Calculation I is not complete for the DFT description of the system being studied. We performed a third self-consistent calculation with the basis set from Calculation II augmented with an appropriate orbital representing an excited level of the ionic species in the system. Again, the occupied energies from Calculations II and III were compared numerically and graphically. This process continued until a given calculation, i.e. N, was found to have the same occupied energy as the two calculations following it, i.e., N + 1 and N + 2. The superposition of the occupied energies from three consecutive calculations is the indication that one has reached the absolute minima of the occupied energies (i.e., the ground state), as required by DFT. A third calculation is needed to confirm the attainment of the absolute minima of the valence bands, as opposed to local minima. The calculation with the smallest basis set of the three calculations provided the DFT description of the material under study and its basis set is called the optimal basis set. For w-BeO, the basis of Calculation II gives these absolute minima of the occupied energies and provides the DFT description of the material.

When the ground state is reached, further calculations with basis sets larger than the optimal basis set do not modify the occupied energies. In other words, if the basis set of Calculation N is the optimal basis set, then Calculation N + 1, N + 2, N + 3, etc. produce the same occupied energies. However, as per the Rayleigh theorem [

Let an eigenvalue equation be is solved with two basis sets containing n and n+1 basis functions, respectively, with the smaller basis set totally included in the larger one. The Rayleigh theorem states that the ordered eigenvalues (from the lowest to the highest) are generate in such a fashion that eigenvalues obtained with (n + 1) functions are lower than or equal to their corresponding values obtained with n functions. This theorem clearly explains the reason very large basis sets containing the optimal one are over-complete for the DFT description of the material being study; these very large basis sets do not change the charge density and the Hamiltonian from their respective values generated with the optimal basis set. With no change in the Hamiltonian, the physics of the problem, a change in eigenvalues (unoccupied ones) is a mathematical artifact and not the expression of a physical interaction contained in the Hamiltonian.

Details of our computational method for a replication of our calculations follow. BeO is an element of the space group P6_{3}mc and have the hexagonal wurtzite structure [

assigned to the special position 2(b) so that Be atoms occupy the sites ( 1 3 , 2 3 , 0 ) and ( 2 3 , 1 3 , 1 2 ) while the oxygen atoms are at ( 1 3 , 2 3 , z ) and ( 2 3 , 1 3 , 1 2 + z ) . In

our self-consistent calculations, we utilized the experimental lattice constants a = 2.6979 Å, c = 4.3772 and z = 0.375 Å at room temperature. Our calculations used Gaussian functions. We employed even-tempered Gaussian functions, with minimum and maximum exponents of 0.3 and 0.5585 × 10^{5} for Be^{2+} and 0.22 and 0.5254 × 10^{5} for O^{2−}. We utilized a 24 k point mesh, with proper weights, in the irreducible Brillouin zone of the wurtzite structure. The computational error for the valence charge was about 0.00479 for 20 electrons or 2 × 10^{−4} per electron. The self-consistent potentials converged to a difference of 10^{−5}.

After the attainment of self-consistency with the optimal basis set, we produced the final bands using 141 k-points in the Brillouin zone. We subsequently utilized 320 k points for the calculations of the total and partial densities of states. We also obtained effectives masses using fits around the minimum and maxima of the conduction and valence bands, respectively.

The DFT description of the electronic properties of w-BeO is provided by the band structure in

Beryllium Be^{2+} (valence electrons) | Oxygen O^{2−} (valence electrons) | Wave functions 2Be^{2+} & 2O^{2−} | Band gap (Γ-Γ) in eV | |
---|---|---|---|---|

1 | 1s^{2}2s^{0}2p^{0} | 2s^{2}2p^{6} | 36 | 10.630 |

2 | 1s^{2}2s^{0}2p^{0}3p^{0} | 2s^{2}2p^{6} | 48 | 10.296 |

3 | 1s^{2}2s^{0}2p^{0}3p^{0}4p^{0} | 2s^{2}2p^{6} | 60 | 10.300 |

4 | 1s^{2}2s^{0}2p^{0}3p^{0}4p^{0}4s^{0} | 2s^{2}2p^{6} | 64 | 9.182 |

5 | 1s^{2}2s^{0}2p^{0}3p^{0}4p^{0} | 2s^{2}2p^{6}3s^{0} | 64 | 7.58 |

6 | 1s^{2}2s^{0}2p^{0}3p^{0}4p^{0} | 2s^{2}2p^{6}3p^{0} | 72 | 9.388 (Indirect) |

unoccupied energies, obtained with basis sets containing the optimal one, is explained by the Raleigh theorem.

The total density (DOS) and partial density (pDOS) of states, obtained from the bands from Calculation II (with the optimal basis set) are given in

Although we do not have experimental results for complete comparison, some of our results agree with measurements by Sashin and coworkers [

We provide, in

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

16.391 | 18.583 | 19.456 | 18.743 | 16.391 | 19.335 |

16.391 | 18.583 | 18.526 | 16.979 | 16.391 | 19.335 |

15.608 | 18.105 | 18.309 | 16.979 | 15.608 | 18.187 |

15.608 | 18.105 | 17.429 | 15.253 | 15.608 | 18.187 |

15.608 | 17.114 | 16.449 | 14.441 | 15.608 | 14.577 |

15.608 | 17.114 | 15.723 | 14.441 | 15.608 | 14.577 |

11.646 | 10.815 | 13.250 | 12.515 | 11.646 | 13.667 |

11.646 | 10.815 | 11.474 | 10.296 | 11.646 | 13.667 |

−0.632 | −1.998 | −1.082 | 0 | −0.632 | −1.574 |

−0.632 | −1.998 | −2.242 | 0 | −0.632 | −1.574 |

−0.632 | −2.107 | −2.946 | −1.187 | −0.632 | −4.646 |

−0.632 | −2.107 | −3.626 | −1.187 | −0.632 | −4.646 |

−3.543 | −5.700 | −4.249 | −1.187 | −3.543 | −4.674 |

−3.543 | −5.700 | −6.039 | −6.446 | −3.543 | −4.674 |

−17.951 | −16.7 | −16.424 | −16.904 | −17.951 | −16.515 |

−17.951 | −16.7 | −16.956 | −18.733 | −17.951 | −16.515 |

further comparison of our findings with some future, experimental data. Such experiments can include optical transitions. X-ray and UV spectroscopy measurements can provide the widths of upper and lower groups of valence bands and the total width of the entire valence band.

Effective masses are used in determining transport properties of a system, i.e., the transport of electrons under the influence of electric fields or carrier gradients. Agreements between measured and calculated electron effective masses show the accuracy of the shape and curvature of the calculated band. Our calculated electron effective mass and those of the light hole and heavy holes 1 and 2, from Γ to A, are respectively 1.298, −0.360, −2.107 to −2.685, and −2.107 to −2.68 m_{o}. We are not aware of measured values of these effective masses for w-BeO.

The content of

Previous ab-initio calculations, with single basis sets, led to stationary solutions. There is an infinite number of stationary solutions. Some do not obtain the ground state, due to deficiency in their basis sets. Others may produce the absolute minima of the occupied energies (the ground state), however, practically all of them have over-complete basis sets that lead to unphysical lowering of unoccupied energies, resulting in an underestimation of the band gaps. The chances for a single basis set calculation to produce the absolute minima of the occupied energies (i.e., the ground state), without using a basis set that is over-complete for the description of the ground state, are practically zero. The use of successively larger basis sets in consecutive, self-consistent calculations is the only way we know 1) to ensure the attainment of the absolute minima of the occupied energies (the ground state) and 2) to avoid unnecessarily large basis sets. Such large basis sets are over-complete and they unphysically lower unoccupied energies to result in band gaps smaller than the actual DFT and experimental ones.

The above points also explain the reason that our estimated band gap is in excellent agreement with experiment, i.e., our computations strictly adhered to conditions inherent to the validity of DFT. The successively larger basis sets led to a minimization far beyond what is attainable with a single basis set following self-consistency iterations. Even though the self-interaction correction (SIC) calculation with a hybrid potential, in

We have described the ground state electronic and related properties of w-BeO by using the BZW-EF method. It ensures the full physical content of calculated quantities by adhering to the pertinent DFT theorem. Unlike previous ab-initio calculations, our calculated energy gap of 10.30 eV is in agreement with experiment. Our calculated density of states (DOS) and partial density of states (p-DOS) are in good agreement with results from electron momentum spectroscopy (EMS). We expect future measurements to confirm our findings. To the best of our knowledge, no measurements of the electron effective masses are available for comparison with our calculated ones.

Our findings for w-BeO illustrate the effectiveness of the BZW-EF method to describe accurately electronic and related properties of semiconductors and insulators, using DFT potentials. We recall that the method: 1) searches and obtains the absolute minima of the occupied energies, and 2) avoids very large basis sets that are over-complete for the description of the ground state. The second DFT theorem requires this search. Hence, future DFT calculations should search for and attain verifiably the absolute minima of the occupied energies of systems under study, and avoid over-complete basis-sets, in order to produce results with the full, physical content of DFT and that agree with experiment.

This research work was funded in part by the US Department of Energy, National, and Nuclear Security Administration (NNSA) (Award Nos. DE-NA0001861 and DE-NA0002630), the National Science Foundation (NSF HRD-1002541), LaSPACE, and LONI-SUBR.

Bamba, C.O., Inakpenu, R., Diakite, Y.I., Franklin, L., Malozovsky, Y., Stewart, A.D. and Bagayoko, D. (2017) Accurate Electronic, Transport, and Related Properties of Wurtzite Beryllium Oxide (w-BeO). Journal of Modern Physics, 8, 1938-1949. https://doi.org/10.4236/jmp.2017.812116