^{1}

^{1}

^{1}

^{2}

^{1}

We report results on electronic, transport, and bulk properties of rock-salt magnesium selenide (MgSe), from density functional theory (DFT) calculations. We utilized a local density approximation (LDA) potential and the linear combination of atomic orbitals formalism (LCAO). We followed the Bagayoko, Zhao, and Williams (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF), to perform a generalized minimization of the energy, down to the actual ground state of the material. We describe the successive, self-consistent calculations, with augmented basis sets, that are needed for this generalized minimization. Due to the generalized minimization, our results have the full, physical content of DFT, as per the second DFT theorem [AIP Advances, 4, 127104 (2014)]. Our calculated, indirect bandgap of 2.49 eV, for a room temperature lattice constant of 5.460 Å, agrees with experimental findings. We present the ground-state band structure, the related total and partial densities of states, DOS and PDOS, respectively, and electron and hole effective masses for the material. Our calculated bulk modulus of 63.1 GPa is in excellent agreement with the experimental value of 62.8 ± 1.6 GPa. Our predicted equilibrium lattice constant, at zero temperature, is 5.424 Å, with a corresponding indirect bandgap of 2.51 eV. We discuss the reasons for the agreements between our findings and available, corresponding, experimental ones, particularly for the band gap, unlike the previous DFT results obtained with ab-initio LDA or GGA potentials.

Group II-VI semiconductors and their alloys have attracted attention from scientific and technological communities, in recent years, due to their various applications. Specifically, magnesium selenium (MgSe) is an attractive binary semiconductor utilized as a key component in II-VI semiconductor alloys having n and p doping abilities [

Rock salt (rs) MgSe has applications in high-temperature and high-power blue and ultraviolet wavelength optics [

Many theoretical studies report an indirect bandgap for MgSe, from Γ to X. Drief et al. [

The presumed, experimental band gap for bulk rock-salt MgSe is 2.47 eV, as reported by Heyd et al. [

Formalism Approach | Potentials | Bandgap Eg (eV) |
---|---|---|

Full Potential Linearized Augmented Plane Wave (FP-LAPW) | LDA | 1.95^{a} |

FP-LAPW | LDA | 1.69^{b} |

Pseudo Potential approximation | LDA | 1.62^{c} |

Full-Potential Linear Muffin-Tin Orbitals (FP-LMTO) | LDA | 3.82^{d} |

Non Local Norm-Conserving Pseudopotentials | LDA | 1.57^{e} |

FP-LAPW | GGA | 1.70^{f} |

FP-LAPW | GGA | 1.77^{g} |

FP-LAPW | GGA | 1.82^{h} |

Perdew-Burke-Ernzerhof (PBE) | GGA | 1.78^{i} |

FP-LMTO | GGA | 1.75^{j} |

Norm-Conserving Pseudo Potentials | GGA-PBE | 1.75^{k} |

FP-LAPW | G_{o}W_{o} | 2.68^{l} |

Tight Binding Linear Muffin-Tin Orbital (TB-LMTO) | LDA | 1.5^{m} |

Pseudopotential Gaussian type | HF | 10.6^{n} |

Linear Combination of Atomic Orbitals-Self Consistent Field (LCAO-SCF) | HF mixes with LDA-PZ | 8.9^{o} |

FP-LAPW | mBJ LDA | 2.53^{p} |

FP-LAPW | mBJ GGA | 2.83^{q} |

Tao-Perdew-Staroverov-Scuseria (TPSS) | mBJ GGA | 2.07^{r} |

^{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 [^{n}Ref [^{o}Ref [^{p}Ref [^{q}Ref [^{r}Ref [

some authors [

Ab-initio LDA potentials led to results that vary from 1.57 eV to 1.95 eV, with an outlier of 3.82 eV. Ab-initio GGA calculations produced band gaps between 1.75 and 1.82 eV. These calculated values, except for the outlier, are much lower than the reported, experimental findings in

Rock Salt Sample Characteristics | Bandgap E_{g} (eV) |
---|---|

Presumed, experimental band gap of rs-MgSe, as reported by Heyd et al. [ | 2.47 |

Nanocrystalline MgSe thin films grown by spray pyrolysis | 2.45^{s} |

MgSe thin films grown by chemical bath deposition | 2.65 - 2.82^{t} |

Nanocrystalline MgSe thin films from spray solution utilizing spray pyrolysis | 2.45 - 2.75^{u} |

Nanocrystalline MgSe thin films deposited by chemical route triethanolamine | 2.65 - 2.79^{v} |

Nanocrystalline MgSe thin films deposited at room temperature by solution growth | 2.49 - 2.7^{w} |

^{s}Ref [^{t}Ref [^{u}Ref [^{v}Ref [^{w}Ref [

gaps of films and nanocrystals, in comparison to that of the bulk, by quantum confinement, it is not expected to be as large as 0.88 eV, the difference between the lowest of the measured gaps (2.45 eV) and the lowest of the calculated ones (1.57 eV). The resolution of this disagreement between experiment and theory is our central motivation for this work. Indeed, the importance of the band gap resides in part in the fact that several properties of materials depend on it, including optical, dielectric, and transport properties. For transport properties, this dependence is partly through the effective masses that are inversely proportional to mobility.

In this work, our implementation of the linear combination of atomic orbitals (LCAO) followed the Bagayoko, Zhao, and Williams (BZW) method [

The computational details that permit the replication of this work follow. Magnesium selenide, in rock-salt structure, is three-dimensional and crystallizes in the cubic F m - 3 ¯ m space group with four (4) formula units of MgSe per unit cell. The Mg atom is located at the edges and on the six faces. Two inter-penetrating face-centered cubic (FCC) lattices are formed with the Se atom. Mg^{2+} ions in the crystal structure are bonded to six (6) equivalent Se^{2}^{−} ions. The positions of the ions of Mg and Se are (0, 0, 0) and (1/2, 1/2, 1/2), respectively.

Our self-consistent calculations were performed utilizing a room temperature, experimental lattice constant of 5.460 Å [^{2+} and Se^{2}^{−}. We employed a set of even-tempered Gaussian exponents to expand the radial parts of the atomic wave functions in terms of the Gaussian functions. The number of even-tempered Gaussian exponents used for the s, p, and d orbitals for Mg^{2+} were 18, 18 and 16, respectively. Likewise, the s, p and d orbitals for Se^{2}^{−}. were 18, 18 and 16, respectively. The maximum and minimum exponents used for Mg^{2+} were 1.1 × 10^{6} and 0.1822, respectively, whereas the corresponding ones for Se^{2}^{−} were 0.24 × 10^{6} and 0.135, respectively. The computational error for the valence charge was 3.289 × 10^{−}^{4} per electron. Self-consistency was reached after 60 iterations, with 81 k-points in the irreducible Broullouin zone, when the difference between the potentials from two consecutive iterations was less than or equal to 10^{−}^{5}.

We list below, in

No. of Calculation | Magnesium(Mg^{2+}) (1s^{2}-Core) | Selenium (Se^{2}^{−}) (1s^{2}2s^{2}2p^{6}-Core) | No. of functions | Band Gap (eV) |
---|---|---|---|---|

I | 2s^{2}2p^{6}3p^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{6} | 40 | 6.20 (Γ-L) |

II | 2s^{2}2p^{6}3p^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{0} | 50 | 6.21 (Γ-L) |

III | 2s^{2}2p^{6}3p^{0}3s^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{0} | 52 | 2.80 (Γ-X) |

IV | 2s^{2}2p^{6}3p^{0}3s^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{0}5p^{0} | 58 | 2.49 (Γ-X) |

V | 2s^{2}2p^{6}3p^{0}3s^{0}4p^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{0}5p^{0} | 64 | 2.50 (Γ-X) |

VI | 2s^{2}2p^{6}3p^{0}3s^{0}4p^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{6}4d^{0}5p^{0}5S^{0}_{ } | 66 | 2.38 (Γ-X) |

L-point | Γ-point | X-point | K-point |
---|---|---|---|

14.654 | 34.882 | 23.504 | 19.386 |

8.936 | 11.259 | 12.389 | 11.965 |

7.403 | 11.259 | 12.389 | 10.665 |

7.349 | 11.259 | 12.293 | 10.267 |

6.314 | 4.372 | 2.492 | 4.005 |

−0.503 | 0.000 | −1.816 | −1.431 |

−0.503 | 0.000 | −1.816 | −3.305 |

−4.975 | 0.000 | −4.281 | −3.748 |

−11.302 | −12.535 | −10.956 | −10.954 |

from Calculation IV, in the energy range of −14 eV to 12 eV. The total valence band width is about 12.61 eV. The width of the lowest laying valence band is 1.75 eV while that of the uppermost group of valence bands is 5.0 eV. The content of

_{f}) set at zero.

The large difference between our calculated band gap and the ones previously obtained with ab initio DFT potentials needs an explanation. The detailed description of our method unveiled the crucial difference between BZW-EF calculations, that perform a generalized minimization of the energy, and other DFT calculations. Indeed, most of the other ab initio DFT calculations employ a single basis set to obtain self-consistent results that are assumed to be those for the ground state. Such results, however, are stationary ones among an infinite number of such solutions [

The Rayleigh theorem [

M_{eXlong} | M_{eXtrans} | Me_{Γ} | M_{hh1} | M_{hh2} | M_{lh} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

(100) | (110) | (111) | (100) | (110) | (111) | (100) | (110) | (111) | |||||

0.488 | 0.283 | 0.317 | 0.939 | 3.268 | 1.471 | 0.939 | 1.064 | 3.268 | 0.265 | 0.231 | 0.187 | ||

size increases, either a given eigenvalue is lowered or it remains the same. Therefore, once the ground state is reached, further augmentation of the basis set does not affect the occupied energies. However, these larger basis sets lower some unoccupied energies, including the lowest laying ones. This extra-lowering of unoccupied energies, a mathematical artifact [

The referenced attainment of the true ground state of materials guarantees that our results possess the full, physical content of DFT. Consequently, our results agree with available, corresponding experimental ones. Bagayoko [_{3}N_{4}, wurtzite InN, cubic InN, and rutile TiO_{2}; these predictions have been confirmed by experiment [

We utilized our well-established BZW-EF computational method to calculate the ground state electronic and related properties of rock salt MgSe. Unlike the previous, ab initio DFT results, our calculated band gap of 2.49 eV agrees with experiment. The same is true for our calculated bulk modulus of 63.1 GPa. Our predictions for the effective masses and hybridization patterns, where the latter are derived from the calculated partial densities of states, are expected to be confirmed by experiment.

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

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

Ayirizia, B.A., Malozovsky, Y., Franklin, L., Bhandari, U. and Bagayoko, D. (2020) Ab-Initio Self-Consistent Density Functional Theory Description of Rock-Salt Magnesium Selenide (MgSe). Materials Sciences and Applications, 11, 401-414. https://doi.org/10.4236/msa.2020.117027