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We report accurate, calculated electronic, transport, and bulk properties of zinc blende gallium arsenide (GaAs). Our ab-initio, non-relativistic, self-con-sistent calculations employed a local density approximation (LDA) potential and the linear combination of atomic orbital (LCAO) formalism. We strictly followed the Bagayoko, Zhao, and William (BZW) method, as enhanced by Ekuma and Franklin (BZW-EF). Our calculated, direct band gap of 1.429 eV, at an experimental lattice constant of 5.65325 Å, is in excellent agreement with the experimental values. The calculated, total density of states data reproduced several experimentally determined peaks. We have predicted an equilibrium lattice constant, a bulk modulus, and a low temperature band gap of 5.632 Å, 75.49 GPa, and 1.520 eV, respectively. The latter two are in excellent agreement with corresponding, experimental values of 75.5 GPa (74.7 GPa) and 1.519 eV, respectively. This work underscores the capability of the local density approximation (LDA) to describe and to predict accurately properties of semiconductors, provided the calculations adhere to the conditions of validity of DFT.

Gallium arsenide is an important electronic and opto-electronic material [

Numerous theoretical results have been reported for the band gap of GaAs. Our focus on the band gap stems from its importance in describing several other properties of semiconductors [

Other results obtained with LDA potentials, as shown in

As shown in

Computational Formalism | Potentials (DFT and others) | a(Å) | Eg (eV) |
---|---|---|---|

LMTO | LDA (fully relativistic local density) | 0.25[a] | |

LDA + V_{w} (with extra potentials) | 1.46[a] | ||

LCGO | LDA | 5.654 | 1.21[b] |

LAPW | LDA | 5.653 | 0.28[c] |

PAW | LDA | 0.330[d] | |

FP -LAPW | LDA | 5.6079 | 0.463[e] |

LDA | 0.28[f] | ||

Self-consistent DFT | LDA-SZ | 5.68 | 0.61[g] |

LDA-SZ-O | 5.66 | 0.78[g] | |

LDA-SZP-O | 5.60 | 0.98[g] | |

LDA-DZ | 5.64 | 0.66[g] | |

LDA-DZP | 5.60 | 0.82[g] | |

LDA-PW | 5.55 | 1.08[h] | |

LDA-PW | 5.55 | 0.7[i] | |

First-principal total-energy calculations | LDA | 1.17[j] | |

First-principal total-energy calculations | LDA | 1.23[k] | |

Plane-wave pseudopotential | LDA | 5.45 | 1.54[l] |

Plane-wave pseudopotential | LDA | 5.654 | 1.04[m] |

FP-LAPW UPP (CASTEP) | LDA | 1.613[n]^{ } | |

LDA-mBJ | 1.46[n]^{ } | ||

LDA-sX | 1.639[n]^{ } | ||

FP-LAPW NCP (SIESTA) | LDA | 0.54755[n]^{ } | |

LDA | 0.23[o]^{ } | ||

LDA | 0.18[p]^{ } | ||

LDA | 0.09[q]^{ } | ||

LDA | 0.32[r]^{ } | ||

GGA | 0.51[f] | ||

GGA-EV | 1.03[f] | ||

GGA-EV | 0.97[s]^{ } | ||

GGA | 0.49[r] | ||

FP-LAPW UPP (CASTEP) | GGA-PBE | 0.329[n] | |

GGA-WC | 0.206[n] | ||

GGA-PBE | 0.52317[n]^{ } | ||

FP-LAPW NCP (SIESTA) | Meta-GGA | 1.27637[n]^{ } | |

Ab initio pseudopotential | GGA | 5.653 | 1.419[t] |

All electron atomic orbit | GGA | 0.82[u] | |
---|---|---|---|

PAW | GGA | 5.734 | 0.674[v] |

PAW | GGA-PBE | 5.648 | 0.43[w] |

FP -LAPW | GGA-WC | 5.6654 | 0.341[e] |

GGA-EV | 0.968[e] | ||

mBJ-LDA | 1.560[e] | ||

mBJ-LDA | 1.64[x] | ||

HSE06 | 1.33[w] | ||

G_{0}W_{0} | 1.51[w] | ||

Plane wave and pseudopotential | GW | 1.133[d] | |

SX | 1.289[d] | ||

Ab initio pseudopotentials | 5.52 | 0.4[y] | |

LUC-INDO | 5.6542 | 1.91[z] | |

EPM | 1.527[α] | ||

EPM | Non local pseudopotential | 1.51[β] | |

EPM | Non local pseudopotential | 5.65 | 1.51[γ] |

Experiments | |||

Experimental | Absorption spectra measurements | 1.519[δ] at low T | |

Photoluminescence measurements | 1.519[ε], low T 1.43[ζ] at 300 K | ||

Magnetoluminescence measurements | 5.65325 | 1.5192[η], low T | |

Transmission measurements | 1.42[θ] at 300 K | ||

Raman measurements | 1.519[ι] at low T 1.425[ι], 300 K | ||

Scanning tunneling microscopy and spectroscopy measurements | 1.7[κ] at 300 K 1.2[λ] at 300 K 1.42[μ] at 300K | ||

Photocapacitance measurements | 1.5[ν] at 77 K |

[a] Ref [

1.519 eV for room and low temperatures, respectively. The calculation that utilized a meta-GGA potential found a gap of 1.276 eV [

The Green function and dressed Coulomb (GW) approximation calculations led to mixed results. The non-self-consistent G_{0}W_{0} calculation obtained a gap of 1.51 eV [

The above overview of the literature points to the need for our work. Indeed, numerous calculated values of the band gap disagree with corresponding, experimental ones. The disagreement between sets of calculated band gaps, as evident above and in

We are aware of some explanations of the failures of many previous calculations to lead to correct values of the band gaps of semiconductors or insulators. Prominent among them are the self- interaction (SI) [

The rest of this paper is organized as follows. This section, devoted to the introduction, is followed by a description of our computational method, in Section 2. We subsequently present our results in Section 3 and discuss them in Section 4. Section 5 provides a short conclusion.

Our calculations are similar to most of the previous ones discussed in

The method searches for the absolute minima of the occupied energies, using successively augmented basis sets, and avoids the destruction of the physical content of the low, unoccupied energies?once the referenced minima are attained. Typically, the implementation starts with a self-consistent calculation that employs a small basis set; this basis set is not to be smaller than the minimum basis set, the one that can just account for all the electrons in the system. A second calculation follows, with a basis set consisting of the previous one plus one additional orbital. The dimension of the Hamiltonian matrix is consequently increased by 2, 6, 10, or 14 for s, p, d, and f orbitals, respectively. Upon the attainment of self-consistency, the occupied energies of Calculation II are compared to those of I, graphically and numerically. In general, upon setting the Fermi level to zero, some occupied energies from Calculation II are found to be lower than corresponding ones from Calculation I. This process of augmenting the basis set and of comparing the occupied energies from a calculation to those of the one immediately preceding it continues until three consecutive calculations lead to the same occupied energies. This criterion is a clear indication of the attainment of the absolute minima of the occupied energies. The first of these three calculations, with the smallest basis set, is the one that provides the DFT description of the material. The basis set for this calculation is the optimal basis set.

While the second of these calculations generally leads to the same occupied and low, unoccupied energies up to 6 - 10 eV, depending on the material, the third of these calculations often lowers some low, unoccupied energies from their values obtained with the optimal basis set. We should note that the referenced three calculations lead to the same electronic charge density. As explained by Bagayoko [

The final implementation of our method, for GaAs, followed a mixture of the BZW and BZW-EF method, as shown below in connection with the tabulation of the successive, self-consistent calculations. In the BZW method, orbitals are added in the order of the increase of the energies of the excited states they represent on the atomic or ionic species in the material. In the BZW-EF method, orbitals are added, on a given atomic or ionic site, as follows: for a given principal quantum number n, the p, d, f polarization orbitals are added before the corresponding, spherically symmetric s orbital for that number. This ordering is based on the fact that, for valence electrons (participating in bonding), polarization has primacy over spherical symmetry.

As discussed below, however, we encountered, for the first time, a special situation where, after adding 4d^{0} orbitals to both Ga^{1+} and As^{1−}, adding the 5p^{0} and 5d^{0} before the 5s^{0} led to an increase of some occupied energies and not just a decrease of some others. It should be noted that the referenced increases do not violate the Rayleigh theorem that is rigorously followed if the Fermi energies are not set to zero. This was the case when 5p^{0} was added for Ga^{1+}; when it was also added for As^{1−}, many occupied energies increased while none decrease; when 5d^{0} was subsequently added for Ga^{1+}, the resulting occupied energies were the same as those of the previous calculation. We recall that many occupied energies from that previous calculation were uniformly higher than those obtained with just the 5p^{0} on both sites. The superscript (0) above signifies a non-occupied orbital. Unlike the above cases, adding the 5s^{0} to the two sites, one at a time, led to the same occupied energies as obtained with the 4d^{0} on both sites. Based on the minimal, occupied energy requirement of DFT, the DFT description of the material was obtained once 4d^{0} was added to both sites. This choice was dictated by the fact that this calculation and the two others following it, with a 5s^{0} orbital on Ga^{1+} and on both ions, produced the same, occupied energies.

Bagayoko [

The following computational details are intended to facilitate the replication of our work. GaAs is III-V semiconductor, with the zinc blende crystal structure in normal conditions of temperature and pressure. We used the experimental, room temperature lattice constant of 5.65325 Å [^{+1} and As^{−1} produced atomic orbitals employed in the solid state calculation. We utilized even-tempered Gaussian exponents, with 0.28 as the minimum and 0.55 × 10^{5} as the maximum, in atomic unit, for Ga^{+1}. We used 18 Gaussian functions for s and p orbitals and 16 for the d orbitals. Similarly, the Gaussian exponents for describing As^{−1} were from 0.2404 to 0.349 × 10^{5}. A mesh size of 60 k points in the irreducible Brillouin zone, with appropriate weights, was used in the iterations for self-consistency. The computational error for the valence charge was about 1.25 × 10^{−3} per electron. The self-consis- tent potentials converged to a difference around 10^{−5} between two consecutive iterations.

With the LDA potential identified above and the computational details, we implemented the LCGO formalism following the BZW-EF method. Upon the attainment of absolute minima of the occupied energies, the optimal basis set was employed to produce the band structure of GaAs. The resulting eigenvalues and corresponding wave functions were utilized to calculate the total (DOS) and partial (pDOS) densities of states, as well as electron and hole effective masses. From the curve of the calculated total energy versus the lattice constant, we obtained the equilibrium lattice constant and the bulk modulus. These results follow below, in Section 3.

We present below the successive calculations that led to the absolute minima of the occupied energies for GaAs. Then, we discuss the electronic energy bands resulting from the calculation with the optimal basis set. We subsequently show the total (DOS) and partial (pDOS) densities of states and effective masses derived from the energy bands. The last results to be discussed pertain to the total energy curve, the equilibrium lattice constant, and the bulk modulus. We show, in

The calculated band structure of GaAs, from Calculation III, is shown in

_{IT}, P_{II}, and P_{III} correspond to the binding energies of 1.0 eV, 6.6 eV, and 11.4eV, respectively. From our calculations, the corresponding values are 1.0 eV, 6.4 eV, and 11.0 eV, respectively. The labels of the peaks are as reported by Ley

Calculation Number | Gallium Orbitals for Ga^{1+} | Orbitals for As^{1−} | No. of Wave Functions | Band Gap in eV |
---|---|---|---|---|

Calc. I | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4} | 52 | 1.380 |

Calc. II | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4} | 62 | 1.368 |

Calc. III | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0 } | 72 | 1.429 |

Calc. IV | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0}5p^{0 } | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0} | 78 | 1.488 |

Calc. V | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0}5p^{0 } | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0}5p^{0 } | 84 | 1.596 |

Calc. VI | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0}5p^{0 } | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0}5p^{0}5d^{0 } | 94 | 1.672 |

Calculation Number | Gallium Orbitals for Ga^{1+} | Orbitals for As^{1−} | No. of Wave Functions | Band Gap in eV |
---|---|---|---|---|

Calc. I | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4} | 52 | 1.380 |

Calc. II | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4} | 62 | 1.368 |

Calc. III | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0} | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0 } | 72 | 1.429 |

Calc. IV | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0}5s^{0 } | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0} | 74 | 1.270 |

Calc. V | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{0}4d^{0}5s^{0 } | 3s^{2}3p^{6}3d^{10}4s^{2}4p^{4}4d^{0}5s^{0 } | 76 | 1.238 |

et al. [

We provide in

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

9.593 | 4.164 | 10.772 | 8.642 |

5.248 | 4.164 | 10.772 | 8.590 |

5.248 | 4.164 | 2.429 | 5.320 |

1.646 | 1.429 | 2.336 | 2.769 |

−1.095 | 0 | −2.572 | −2.150 |

−1.095 | 0 | −2.572 | −3.601 |

−6.370 | 0 | −6.582 | −6.379 |

−10.697 | −12.439 | −9.945 | −9.987 |

−15.710 | −15.711 | −15.700 | −15.705 |

−15.710 | −15.711 | −15.731 | −15.722 |

−15.802 | −15.821 | −15.778 | −15.783 |

−15.802 | −15.821 | −15.778 | −15.786 |

−15.870 | −15.821 | −15.903 | −15.891 |

tal measurements from X-ray, ultra violet (UV) or other spectroscopies. From this content, the widths of the upper most, middle, and lower most groups of valence bands are 6.58 eV, 2.494 eV, and 0.203 eV, respectively. The total width of the valence band is 15.903 eV.

We calculated the effective masses of n-type carriers for GaAs, using the electronic structure from Calculation III (in

Column 2 of _{e}) and at the top of the valence bands (m_{hh} and m_{lh}) at the Г point. These effective masses are provided in all three relevant directions, as indicated in Column 1. The values in the three directions permit the determination, at a glance, of the isotropic or anisotropic nature of the effective mass at the point. The importance of accurate effective masses resides in part in the fact that they are inversely proportional to the drift velocity, field current, and mobility of the corresponding charges.

Our calculations, as shown in

Our Work | Theo [a] EPM | Theo [b] | Theo [c]^{ } | Theo [d]^{ } | Expt [e] Room T | Expt [f] Room T | |
---|---|---|---|---|---|---|---|

m_{e} (Г-L) | 0.077 | 0.066 | 0.012 | 0.070 | 0.063 | 0.0635 | |

m_{e} (Г-X) | 0.077 | 0.030 | |||||

m_{e} (Г-K) | 0.078 | ||||||

m_{hh} (Г-L) | 0.865 | 0.866 | 0.827 | 0.50 | 0.643 | ||

m_{hh} (Г-X) | 0.359 | 0.342 | 0.334 | 0.320 | |||

m_{hh} (Г-K) | 0.516 | ||||||

m_{lh} (Г-L) | 0.062 | 0.056 | 0.076 | 0.081 | |||

m_{lh} (Г-X) | 0.076 | 0.093 | 0.068 | 0.036 | |||

m_{lh} (Г-K) | 0.070 |

[a] Reference [

From our overview of the literature and the content of

Our explanation of the excellent agreements noted above rests on the fact that our calculations, with the BZW-EF method, strictly adhered to necessary conditions [

We performed ab-initio, self-consistent calculations of electronic, transport, and bulk properties of GaAs. Our results, unlike those of many previous ab-initio calculations, agree very well with experiment, for the band gaps, the total density of states, and the bulk modulus; they also agree with experiment for the effective masses, where the latter are inversely related to the mobility of charge carriers. We credit our strict adherence to conditions of validity for DFT or LDA potentials, with our implementation of the BZW-EF method, for the above agreements between our calculated results and experimental ones.

This research was funded by the Malian Ministry of Higher Education and Scientific Research, through the Training of Trainers Program (TTP), the US National Science Foundation [NSF, Award Nos. EPS-1003897, NSF (2010-2015)- RII-SUBR, and HRD-1002541], the US Department of Energy, National Nuclear Security Administration (NNSA, Award No. DE-NA0002630), and LONI-SUBR.

Diakite, Y.I., Traore, S.D., Malozovsky, Y., Khamala, B., Franklin, L. and Bagayoko, D. (2017) Accurate Electronic, Transport, and Bulk Properties of Zinc Blende Gallium Arsenide (Zb-GaAs). Journal of Modern Physics, 8, 531-546. https://doi.org/10.4236/jmp.2017.84035