Electronic Structure of AlAs Nanocrystals

Ab initio restricted Hartree-Fock method within the framework of large unit cell formalism is used to investigate the electronic structure of the core and oxid ized surface of AlAs nanocrystals. Large unit cells of 8, 16, 54, 64 and 128 atoms are used in the present analysis. Calcu lations are performed utilizing the Gaussian (03) software. The investigated properties include the cohesive energy, energy gap, valence bandwidth, conduction bandwidth and the density of states of the energy levels. Interesting results are obtained which reveal that the electronic structure of AlAs nanocrystals differs significantly from that of the bulk AlAs crystal. A lso, it is found that the energy gap, valence band width and cohesive energy (absolute value) increase as the AlAs large unit cell size increases, for the core part. Whereas, the energy gap of oxygenated (001) (1×1) surface decreases with increasing the large unit cell size. The energy gap is controlled by the surface part of the nanocrystal. The surface part has lower symmetry than the core part with smaller energy gap and wider valence bandwidth. The density of states of the core part is h igher than that of the surface part . Th is is due to the broken bonds and the discontinuity at the surface and the existence of new kind of atoms (oxygen atoms). The present method has threefold results: the method can be used to obtain the converged electronic structure of bulk, surface, and nanocrystals.


Introduction
Aluminiu m arsenide (AlAs) is a co mpound semiconductor with an indirect band gap and it has a zinc-b lend (zb) structure [1]. AlAs is the most technologically important compound semiconductor material, where it has many optoelectronics and photovoltaic applications in space solar cells, telecommun ication and data communicat ion networks, mobiles, telephones, laptop computers, compact disc players, etc [2]. Therefore, alu min iu m arsenide has been intensively investigated in the recent years, where Bouarissa and Boucenna [3] studied the electronic, optical and mechanical properties of AlAs and they showed that the results are in agreement with the availab le experimental and theoretical data. Chimata [4] studied theoretically the electronic structure of alu miniu m arsenide crystal, and they found that the band gap value obtained is in good agreement with the experimental value. Shimazaki and Asai[5] studied the energy band structure of Si, AlP, AlAs, Ga P and GaAs using screened Hartree-Fock exchange method, and they found that the direct and indirect band gaps don't agree with experimental values.
The afo rement ioned researches and most o f oth er researches deal with bulk A lAs crystals. Due to our extended literature survey, it is found that the electronic properties of AlAs nanocrystals are not investigated comprehensively. So the aim of the present work is to achieve this task. In the present work we introduce the ab-initio restricted Hartree-Fock(RHF) formalis m within the frame of large unit cell (LUC) method to investigate the electronic structure of AlAs nanocrystals (nc s ) employing Gaussian (03) software [6] to perform the calculations. This method has been chosen in the present work rather than other methods because this method is very reliab le to analyse the electronic properties of solids, and because of the significant advantage of ab-init io methods which is the ability to study reactions that involve breaking or formation of covalent bonds, which correspond to multiple electronic states. The downside of ab-in itio methods is their co mputational cost. They often take enormous amounts of computer t ime, memory and disk space [7].

Theory
The main idea o f the large unit cell is co mputing the electronic structure of the unit cell extended in a special manner at k=0 in the reduced Brillouin zone [8,9]. Using the linear combination of ato mic orb itals (LCAO), the crystal wavefunction in the (LUC-RHF) fo rmalis m is written in the following form [10,11] where ψ i is the molecular orb ital, C μi are the mo lecular orbital expansion coefficients, µ ϕ is the atomic orb ital, N 0 is the number of the atomic orbitals. The total electronic energy can be written as [12,13,14]: where H ii is the expectation value of the one electron core Hamiltonian corresponding to the molecu lar orb ital ( i ψ ), J ij and K ij are the Coulomb and exchange operators respectively, and they are expressed as follows: Substituting the linear expansion of Eq.(1) in the molecular orbital integrals, yields: where H µν is the core Hamiltonian matrix elements, µν λσ is the differential overlap matrix elements, 1 2 12 The total electronic energy of Eq.(2) can be written in terms of integral over ato mic orbitals. By substituting the previous expressions in the equation of the electronic energy, one can obtain 1 1 where P µν is the density matrix elements, The summation is over occupied orbitals only for closed shell systems which is the case in the present work. Applying the variational method to Eq.(11), a small variat ion of the mo lecular orbital i ψ can now be given as: And when the condition for a stationary point ( 0) δε = is applied, one can obtain the follo wing final form [15,16]: where F µν is the Fock mat rix,

Results and Discussion
In the present work we have used Gaussian (3) software to calculate the lattice constant, energy gap, cohesive energy, valence band width and degeneracy of states of AlAs nc s (core and surface). These properties are illustrated in table (1) and they are compared with other experimental and theoretical results in table (2). We d ivided the work into two parts, core and surface parts. In the core part we chose the number of atoms (8,16, 54, 64, 128) fo r the (LUC), wh ile in the surface part the choice of nu mber of atoms is (8 and 64) because these numbers of atoms give cubic shapes. The calculated total energy o f A lAs nc s as a function of lattice constant for the LUC of (8) ato ms is shown in Fig. (1). The equilib riu m lattice constant is at the minimu m of the total energy curve.  Fro m Fig. (1), it is shown that the total energy in the core part decreases with increasing the lattice constant till it reaches the equilib riu m lattice constant. After the equilibriu m lattice constant, the total energy increases with the increase of lattice constant. The behaviour of the total energy as a function of the lattice constant is due to the attraction forces that take place at the large distances between atoms. Decreasing the distance between atoms leads to decreasing the total energy. At shorter distances between atoms, the repulsion forces become dominant and take place to cause the increase of the total energy. The stability of the nanocrystal at the equilib riu m lattice constant is due to the equality of the attraction and repulsion forces at this point. In the surface part, we show that the total energy has the same behaviour as that of the core with less values of the total energy of the surface compared with that of the core. The decrease of the total energy for surface is caused by broken bonds and discontinuity at the surface leading to decrease in lattice constant. Also we show that the cohesive energy and valence band width increase with increasing the number of atoms per (LUC), reach ing to 64 ato ms in which they tend to stabilize, with the increasing number of atoms as shown in Fig. (2) and Fig. (3) respectively. The energy gap increases with increasing the number of atoms for core as in Fig. (4). Also the energy gap of oxygenated(001)-(1×1) surface increases with increasing the nu mber of atoms. The conduction band width has some fluctuations.
An interesting observation can be deduced from Table (2), which is the significant difference between the electronic properties of AlAs nanocrystals compared with the corresponding properties of bulk AlAs crystal.
The cohesive energy of AlAs (nc s ) has high value in comparison with the experimental and theoretical values.
Many factors that affect the cohesive energy are not introduced in the present calculations that include: correlation corrections, relativ istic corrections, in addition of Simp le STO-3G basis is used in the present work to be able to reach higher number of core and surface ato ms. The p resent value of the cohesive energy and other properties can be greatly enhanced by merely changing the simp le STO-3G basis used in the present calculations to a mo re sophisticated basis states. However, more sophisticated basis states consume more memo ry and co mputer t ime and will eventually p revent us from reaching large number o f atoms. The high value of the band gap can be attributed to the approximations involved in HF method as well as in the present RHF-LUC forma lis m. and due to bonds nature (mixed bonds) in AlAs (nc s ), this sometimes causes very large band gaps for these nanocrystals [21].   The density of states of AlAs (nc s ) as a function of the energy levels was calculated fo r both core and surface parts. Fig. (5a) shows the density of states for (8) ato ms core LUC and Fig. (5b) represents the density of states for o xygenated (001)-(1×1) AlAs (nc s ) surface. The difference of the degeneracy of the surface in comparison with that of the core is caused by the broken bonds and the discontinuity at the surface and the existence of new kind of atoms (o xygen atoms) that leads to varying the bond lengths and angles, and then to changing the geometry of the structure.

Conclusions
1. The present work reveals an interesting fact that the electronic structure of AlAs nanocrystals differs significantly fro m that of the bulk AlAs crystal.
2. The present method cannot be used only to investigate the electronic properties of bulk crystals, but it may be also used to investigate the electronic properties of nanocrystals.
3. The lattice constant of all size of AlAs nc s (core and surface) decreases with increasing the number of atoms in the LUC.
4. The present calculations show that the energy gap and valence band (absolute value) increase as the AlAs (nc s ) LUC size increases for both the core and oxygenated (001)-(1×1) surface. Also the cohesive energy (absolute value) for the core part increase with increasing the number of atoms.
5. The energy gap is controlled by the surface part of the nanocrystal. The surface part has lo wer symmetry than the core part with s maller energy gap and wider valence band. The density of states of the core part is higher than that of the surface part. This reflects the high symmetry and equal bond lengths and angles in the perfect AlAs nanocrystals structure.