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This research paper is on Density Functional Theory (DFT) within Local Density Approximation. The calculation was performed using Fritz Haber Institute Ab-initio Molecular Simulations (FHIAIMS) code based on numerical atomic-centered orbital basis sets. The electronic band structure, total density of state (DOS) and band gap energy were calculated for Gallium-Arsenide and Aluminium-Arsenide in diamond structures. The result of minimum total energy and computational time obtained from the experimental lattice constant 5.63 A for both Gallium Arsenide and Aluminium Arsenide is -114,915.7903 eV and 64.989 s, respectively. The electronic band structure analysis shows that Aluminium-Arsenide is an indirect band gap semiconductor while Gallium-Arsenide is a direct band gap semiconductor. The energy gap results obtained for GaAs is 0.37 eV and AlAs is 1.42 eV. The band gap in GaAs observed is very small when compared to AlAs. This indicates that GaAs can exhibit high transport property of the electron in the semiconductor which makes it suitable for optoelectronics devices while the wider band gap of AlAs indicates their potentials can be used in high temperature and strong electric fields device applications. The results reveal a good agreement within reasonable acceptable errors when compared with the theoretical and experimental values obtained in the work of Federico and Yin wang [1] [2].

The understanding of the physical properties of interacting many body systems is one of the most important goals of physics after the foundation of quantum mechanics in the mid 1920’s (Rerum, 2005) [

Perdew shows that when DFT is extended to fractional occupation number, the exchange-correlation to fractional occupation number of the electron count is discontinuous at the Fermi level. This fundamental discontinuity in the exact Vxc is precisely the difference between the kohn-sham and true band gaps, but is not reproduced in LDA or GGA. Another major contribution to the band-gap error arises from the electrostatic electron-electron contribution to the Hamiltonian which is usually computed as the Hartree energy E_{H}. Although the equation includes the correct coulomb repulsion, by using the total density, it has included a coulomb repulsion between an electron and its own charge. This spurious self-interaction is exactly cancelled by the exchange term in some non-DFT methods but is only partially cancelled by LDA (or GGA) exchange. This residual self-interaction is one of the most significant causes of the under estimation of the band gap in LDA or GGA based DFT calculations. [

In Lichteinstein work, we observed the introduction of Hubbard U to correcting the self-interaction as LDA + U or DFT + U method; this phenomenon introduces a repulsion between the localized electrons on a given atom [

FHI-aims code solves the DFT + U issues by treating all electrons in an equivalent way. In some special cases of different element, frozen core treatment are applied where one compute the correlation energy of only the valence but not the core electron in second order Moller-Plessett (MP2) perturbation theory and for any two-elec- tron coulomb operator (hybrid functional, Hatree-forck, MP2, or RPA perturbation theory, GW correction etc.) therefore auxillary basis is used to expand the coulomb matrix (four basis functions ≡O(N^{4}) matrix elements) into a two-center coulomb matrix, leading instead to O(N^{3}) additional overlap matrix elements which offers an ad-hoc correction for strongly correlated systems at negligible computational cost.

FHI-aim also allows fixing the mixing factor between Around Mean Field (AMF) and Fully-localized limits (FLL). These two common schemes deal comfortably with the double counting problem in DFT + U. The AMF method assumes that the effect of the DFT + U term on the actual occupations remains small, so that the occupation can be assumed to be equal with each shell for the purpose of the double counting corrections while the FLL method assumes a maximal effect of the DFT + U term on the occupation numbers, handling double counting correctly in the case that all orbitals within the shell are either fully occupied or empty. This improves the handling of intermediate range in self-consistent mixing of both limits [

This work therefore attempt to calculate and estimate the electronic energy band structures and density of state of Gallium-Arsenide and Aluminum-Arsenide semiconductor from ground state density using DFT computational code FHI-aims. The experimental lattice constant parameter values was used to calculate the minimum total energy and tested for different k-grids in order to determine the minimum time required for energy stability of the semiconductor in diamond structure.

The first principles of H-K theorem demonstrates that the ground state properties of a many-electron system are uniquely determined by an electron density that depends on only three spatial co-ordinates which reduces our problem to 3 spatial co-ordinates from 3 N spatial co-ordinates for N body problem because of the use of density functional. The N particle system of interacting particles with 3 N degrees of freedom is reduced to a significantly more tractable problem, which deals with a function (density) of only three variables. The many-body effects incorporated in the exchange-correlation potential are typically approximated within either the local density approximation or the generalized gradient approximation. The formulation applies to any system of interacting particles in an external potential

The first term in this equation corresponds to the kinetic energy of the interacting electrons, the second term is the external potential acting on the electrons due to the ions, the third term is the electron Coulomb interaction, and the last term is the interaction energy of the nuclei. Since the Hamiltonian is thus fully determined (except for a constant shift of the energy), it follows that all properties of the system can be found given only the ground

state density

The minimization of energy functional

where ψ_{i}’s are the single-particle wavefunctions for the non-interacting electron gas with ground state charge density

The

where the first term corresponds to the kinetic energy of a non-interacting electron gas at the same density_{i}’s subject to the orthonormality constraint, the following set of Schrodinger equations is obtained

where the effective potential

Equations ((5) and (6)) are called the Kohn-Sham equations and have to be solved self-consistently because of the dependence of

and

with

FHI-aims (“Fritz Haber Institute ab-initio molecular simulations”) Code [

The focus of this work is to use DFT to estimate the local density approximations (LDA) in exchange-corre- lation potential of Ceperley and Alder work as parametrized by Perdew and Zunger [

To compute the band structure of GaAs, and AlAs, we first calculate the ground state total energies of the most stable structure of these semiconductors as a function of their lattice constants [

All calculations were carried out using FHI-aims code upgrade 6 (released on 17th July, 2011; version 071711_6). It works on any Linux based operating system. Computations can only be carried out after building an executable binary file since the FHI-aims package is distributed in a source code form.

A working Linux-based operating system (Ubuntu 14.04LTS is used in this case), A working FORTRAN 95 compiler, in this case we use intel’s ifort compiler (specifically Composerxe 2013_sp1.3.174) was installed and used for computation in this work. A compiled version of lapack library, and a library providing optimized linear algebra subroutines (BLAS). Standard libraries such as Intel’s mkl or IBM’s essl provide both lapack and BLAS support. Intel’s Composerxe 2013_sp1.3.174 comes with mkl. All necessary adjustment were made for building the executable binary file for running the code and the executable program was successfully built.

The FHI-aims requires two input files: the control.in which contains all runtime-specific information and the geometry.in which contains information directly related to the atomic structure for a given calculation. The two input files must be placed in the same directory from where the FHI-aims binary file is invoked at the terminal.

Our first step towards studying periodic systems with FHI-aims is to construct periodic geometries in the FHI-aims geometry input format (geometry.in). Next, we set basic parameters in control.in for periodic calculations. Finally, we compare total energies of different GaAs, and AlAs bulk geometries.

Geometry.in files for the GaAs and AlAs structures were constructed varying the lattice constants around the experimental lattice constants a of 5.63 Å for GaAs and 5.63 Å for AlAs. At each lattice constant, if the symmetry of the system allows the ions to move, a separate geometric optimization must be performed. The form of total energy as a function of lattice constant is asymmetric and is well described by Murnaghan’s equation [

In setting up the geometry.in file of a periodic structure in FHI-aims, the lattice vectors of the two semiconductors as well as their atomic positions in the unit cell are specified.

The electronic band structure of the stable phases of the semiconductor were then calculated along the high symmetry lines of the Brillouin zone by computing the control.in settings to calculate the band-structures and density of state. This was done when the control.in input files for GaAs and AlAs were created with the appropriate settings.

Thus the calculation is performed as follows:

# Geometry for GaAs

lattice_vector 0.00000000 2.82500000 2.82500000

lattice_vector 2.82500000 0.00000000 2.82500000

lattice_vector 2.82500000 2.82500000 0.00000000

atom_frac 0.00000000 0.00000000 0.00000000 Ga

atom_frac 0.25000000 0.25000000 0.25000000 As

# Geometry for AlAs

lattice_vector 0.00000000 2.83000000 2.83000000

lattice_vector 2.83000000 0.00000000 2.83000000

lattice_vector 2.83000000 2.83000000 0.00000000

atom_frac 0.00000000 0.00000000 0.00000000 Al

atom_frac 0.25000000 0.25000000 0.25000000 As

while the control.in input files for the band structure of GaAs and AlAs were created with the following settings

control.in for GaAs

# Physical model

xc pw-lda

spin none

relativistic atomic_zora scalar

# SCF convergence

sc_accuracy_rho 1E-5

sc_accuracy_eev 1E−3

sc_accuracy_etot 1E−6

sc_iter_limit 100

# k-grid

k_grid 12 12 12

# Density of states

output dos -18 0 1000 0.1

dos_kgrid_factors 5 5 5

# High-symmetry k-points for diamond bandstructure output

output band 0.5 0.5 0.5 0.0 0.0 0.0 50 L Gamma

output band 0.0 0.0 0.0 0.0 0.5 0.5 50 Gamma X

output band 0.0 0.5 0.5 0.25 0.5 0.75 50 X W

output band 0.25 0.5 0.75 0.375 0.375 0.75 50 W K

control.in for AlAs

# Physical model

xc pw-lda

spin none

relativistic atomic_zora scalar

# SCF convergence

sc_accuracy_rho 1E−5

sc_accuracy_eev 1E−3

sc_accuracy_etot 1E−6

sc_iter_limit 100

# k-grid

k_grid 12 12 12

# Density of states

output dos -18 0 1000 0.1

dos_kgrid_factors 5 5 5

# High-symmetry k-points for diamond bandstructure output

output band 0.5 0.5 0.5 0.0 0.0 0.0 50 L Gamma

output band 0.0 0.0 0.0 0.0 0.5 0.5 50 Gamma X

output band 0.0 0.5 0.5 0.25 0.5 0.75 50 X W

output band 0.25 0.5 0.75 0.375 0.375 0.75 50 W K

The band structures of GaAs and AlAs were calculated and the aimsplot.py was used to plot the band structures. The position of the Fermi level in the band structure of these crystals is shown by the zero on the energy scale and that of symmetry points are indicated by vertical lines on the band graph in

more electronegative element in the solid, while those near the conduction band minimum have p anti-bonding character and are associated with less electronegativity. The energy band gap shown in

The next region of noticed is a peak arising from the onset of the second valence band which shows there is no energy variation along the symmetry direction; in fact, it is very flat over the entire square face of the Brillouin zone. The energy band configuration results in a sharp onset of states above the antisymmetric gap. The character of state associated with the second valence band changes from predominantly cation s-like states at the bond edge to predominantly anion p-like state at the band maximum. The third region of interest in the density of state extends from the onset of the third valence band (at about 4 eV below the valence band maximum) to the valence band maximum. This region encompasses the top two valence bands and is predominantly p-like and is associated with anion state.

Showing the values of

Material | Bands | Symmetry points (eV) | Energy Gap (eV) | Fermi Energy | Max/Min Energy band | ||||
---|---|---|---|---|---|---|---|---|---|

Gallium-Arsenide (GaAs) | Valence Band | overlapping | 0.367 | −0.100682 | |||||

L | −11.35070 | −6.98844 | −1.43231 | ||||||

Γ | −13.09560 | −0.28436 | −0.284356 | ||||||

X | −10.57010 | −7.17211 | −2.94762 | ||||||

W | −10.52420 | −6.98844 | −3.86599 Py −3.63040 Pz | ||||||

Conduction Band | L | 0.58809 | 4.35340 | ||||||

Γ | 3.48095 | 0.0829912 | |||||||

X | 1.09320 | 1.32279 |

Material | Bands | Symmetry points (eV) | Energy Gap (eV) | Fermi Energy | Max/Min Energy band | ||||
---|---|---|---|---|---|---|---|---|---|

Aluminium-Arsenide (AlAs) | Valence Band | overlapping | 1.4235 | 0.093005 | |||||

−11.89400 | −6.98076 | −2.20525 | |||||||

−13.27160 | −1.37872 | −1.37872 | −1.37872 | −1.37872 | |||||

−11.34300 | −6.84300 | −3.53688 | |||||||

−11.29710 | −6.42974 | −4.59300 P_{y} −4.22566 P_{z} | |||||||

Conduction Band | 0.68761 | 3.30495 | |||||||

0.54985 | 2.89169 | ||||||||

0.87128 | 0.04475 | 0.04475 |

Solid | Method in this work | E_{g}(eV) in this work | E_{g}(eV) Theoretical value [ | E_{g}(eV) Theoretical value [ | Expt. values [ | Expt. values [ |
---|---|---|---|---|---|---|

Gallium-Arsenide (GaAs) | LDA | 0.37 | 0.49 | 0.67 | 1.63 | 1.52 |

Aluminium-Arsenide (AlAs) | LDA | 1.42 | 2.01 | 1.37 | 2.32 | 3.09 |

However, the under estimation of the band gap from the experimental is mainly due to the fact that the exact functional in the Hohenberg-Kohn theorem is not known. Therefore the comparison of FHI-AIMS approximations for the exchange correlation predicts accurately the band gap when compared to other theoretical values [

This work has successfully employed Density Functional Theory method to calculate and estimate the band structure of Gallium-Arsenide (GaAs) and Aluminium-Arsenide (AlAs) using FHI-AIMS which was successfully installed and the knowledge of the input parameters which include the geometry.in and the control.in was carefully optimized for the band structure studies.

All calculations were carried out using FHI-aims code upgrade 6 (released on 17th July, 2011; version 071711_6) which works on Linux based operating system. In the calculation, Local Density Approximation (LDA) has been used to approximate the exchange correlation energy which varied the treatment of exchange correlation (LDA) to Kohn-Sham DFT leaving all other settings constant.

The experimentally lattice constant parameter value was used to calculate the minimum total energy and tested for different k-grids. The minimum total energy obtained from the experimental lattice constant of Gallium Arsenide 5.63 A and Aluminium Arsenide 5.63 A results in −114915.7903 eV for GaAs and AlAs with a computational time of 64.989 s, for the semiconductors. The result obtained shows that 12 × 12 × 12 k-grids enables the energy stability of the semiconductors in diamond structure with less computational time.

The calculated electronic band structure results shows that AlAs is an indirect band gap semiconductor with 1.42 eV while GaAs is a direct semiconductor with energy band gap of 0.37 eV. This shows significant improvement compared to other theoretical calculation obtained.

The calculated band width in this work shows large values of 13.00 eV and 13.27 eV for GaAs and AlAs and there is no interaction between the s and p orbital of GaAs and AlAs to form hybridization this gives the semiconductor an improved ionic characteristic nature.

The DOS energy level within the semiconductors shows considerable high state of electron occupation and the DOS observed around the Fermi level for the semiconductors at zero level indicate that they have conducting properties.

In general, FHI-AIMS code has shown better accuracy and prediction of band structure calculation within a reasonable computational time when compared to some other DFT theoretical programs observed in literature.

J. A. Owolabi,M. Y. Onimisi,S. G. Abdu,G. O. Olowomofe, (2016) Determination of Band Structure of Gallium-Arsenide and Aluminium-Arsenide Using Density Functional Theory. Computational Chemistry,04,73-82. doi: 10.4236/cc.2016.43007