## Abstract

First principles calculations are performed for the perfect GaAs crystal, the double Ga vacancies (V_{Ga})_{2}, and the ternary complex defect (As_{Ga}V_{As}V_{Ga}), using the state-of-the-art computational method with the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional to correct the band gap and account for a proper description of the interaction between defects states and bulk states. Three shallow acceptor defect levels are found due to the creation of (V_{Ga})_{2} with nearest-neighbor As dangling bonds. However, for GaAs with the ternary complex defects (As_{Ga}V_{As}V_{Ga}), the As antisite As_{Ga} and the V_{As}’s nearest-neighbor Ga dangling bonds provoke several donor defect states. The lowest donor defect state locates at 0.85 eV below the bottom of conduction band, which is very close to the experimental observation of the EL2 defect level. In addition, structual evolution from (V_{Ga})_{2} defect to the ternary defect complex (As_{Ga}V_{As}V_{Ga}) is simulated by *ab initio* molecular dynamic (MD) calculation at different temperatures. The MD results demonstrate that the ternary complex defect (As_{Ga}V_{As}V_{Ga}) can be converted from the double Ga vacancies (V_{Ga})_{2} at room temperature, and it can exist stably at higher temperature. The present work is helpful to unravel the microstructure and the forming mechanism of the EL2 defect, to find out methods to improve the performance of the GaAs saturable absorber by changing the growth conditions of GaAs crystal.

©2012 Optical Society of America

## 1. Introduciton

Semiconductor saturable absorber Q-switched all-solid-state lasers are desirable for many potential applications in remote sensing, ranging, micromachining, and nonlinear wavelength conversion. Compared to other saturable absorbers, GaAs saturable absorber has the advantages of stable photochemical property and saturable absorption, good thermal conductivity, no degradation and high damage threshold [1–4]. However, since the photon energy at 1.06 μm wavelength is far below the GaAs band gap of 1.43 eV, the absorption at this wavelength in the GaAs saturable absorber is believed to be due to the deep donor EL2 defect located 0.82 eV beneath the bottom of the conduction band [5]. To make GaAs act as a passive Q-switch as a result of this saturable absorption for 1.06 μm wavelength laser, it is essential to increase the amount of EL2 defects in GaAs saturable absorber to design the non-linear loss, linear loss, recovery time and the modulation depth, in contrast to the fact that the concentration of EL2 deep-level defects is expected to be low for optoelectronic or high-frequency electronic devices applications. Therefore, the understanding of the microstructure and the formation mechanism of the EL2 defect is critical to control the concentration of EL2 defects through improving the growth conditions of GaAs crystal.

Up to present, the characteristics and forming details of EL2 deep-level defect have been extensively studied from both the experimental and theoretical aspects [6–10], and many models of the stable configuration of the EL2 deep level defect have been proposed, e.g. isolated As_{Ga} antisite, binary complex (As_{Ga}V_{As}, As_{Ga}V_{Ga}, V_{As}V_{Ga}, As_{Ga}As_{i}), ternary complex (V_{Ga}As_{Ga}V_{Ga}, As_{Ga}V_{As}V_{Ga}, As_{Ga}V_{Ga}V_{As}), multicomponent complex, and arsenic cluster. However, the origin of the EL2 defect in GaAs has not yet been identified. Current questions focus on whether the EL2 defect originates from the isolated arsenic antisite As_{Ga} or from some complex involving As_{Ga} as a core, and if the latter is true, what are the nature and the configuration with its partners.

Since the double Ga vacancies (V_{Ga})_{2} is the dominant vacancy species in the GaAs crystal grown from an arsenic-rich circumstance, and it is very likely to convert to the ternary complex As_{Ga}V_{As}V_{Ga} [11], our work is aimed to investigate the relationship between EL2 defect and the ternary complex As_{Ga}V_{As}V_{Ga}, and the possible formatting process of a EL2 defect from a double Ga vacancy (V_{Ga})_{2}.

To the best of our knowledge, there have been few first principles investigations to the EL2 defect. Since density functional theory (DFT) using the local density approximation (LDA) or the generalized gradient approximation (GGA) often severely underestimates the band gap and overestimates the lattice constants, it is much better to apply a hybrid functional in the first principles calculations. This usually opens the band gap, and excellent agreement between theory and experiment is found for the band gap to most semiconductors [12], which also can apply to GaAs.

In this paper, by first principles calculations based on DFT as implemented in the Vienna *ab initio* simulation package (VASP) [13, 14] code with a modified Heyd-Scuseria-Ernzerhof (HSE) hybrid exchange-correlation functional, the electronic structure and the band gaps of perfect GaAs material and GaAs with defects have been more accurately obtained, and positions of the defect levels within the forbidden band have been located exactly. The intercomparison with the experimental results shows that a deep donor defect level is formed by the ternary complex As_{Ga}V_{As}V_{Ga} in GaAs, which is very close to the EL2 defect level. Then we have looked into the evolution process of the double Ga vacancies to the ternary complex As_{Ga}V_{As}V_{Ga} (EL2 defect) by *ab initio* molecular dynamic (MD) simulation at different temperatures, in order to obtain a detailed understanding of the micromechanism of the EL2 deep-level in GaAs. It is found that the ternary complex As_{Ga}V_{As}V_{Ga} can be converted from the double Ga vacancies (V_{Ga})_{2} at room temperature, and can exist stably at higher temperature.

## 2. Calculations of band structure and densities of states (DOS)

To model the defects, a super-cell approach is used, in which defects are repeated periodically. We use super-cells of 64 atoms, which is a reasonable compromise between efficiency and accuracy. Taking two adjacent Ga atoms from the perfect super-cell, we can build the model of the double Ga vacancies (V_{Ga})_{2} as shown in Fig. 1
.

The model of the ternary complex As_{Ga}V_{As}V_{Ga} is plotted in Fig. 2
, in which a V_{As} is in the nearest neighborhood of the arsenic antisite As_{Ga} and V_{Ga} is in the next-nearest neighborhood of As_{Ga}.

For our DFT calculations, the VASP code are used with the plane-wave projector augmented-wave (PAW) [15, 16] pseudopotentials, which reconstruct the exact valence wave functions with all nodes in the core region, adopting smaller core radii than the radii used for the ultra-soft pseudopotentials. Therefore, the PAW potentials are more accurate than the ultra-soft pseudopotentials, and the size of the basis-set can be kept very small. The outermost s- and p-electrons of the As atom, the outermost s-, p- and d-electrons of the Ga atom are treated as valence electrons. The atomic wave functions are expanded into plane-waves up to a cutoff energy 300 eV. Moreover, we applied the semilocal Perdew-Burke-Ernzerhof (PBE) [17] exchange-correlation functional and the Heyd-Scuseria- Ernzerhof (HSE) [18] hybrid functional, which can overcome the band-gap problem in order to correctly describe the electronic and structural properties of GaAs materials. Hybrid functional mixes about 25% nonlocal Hartree-Fock exchange with 75% semilocal exchange, and the HSE screening parameter is set to a value of 0.2 Å^{−1} [19]. Brillouin-zone integration is performed on a Г-centered symmetry 2 × 2 × 2 mesh using a small Gaussian broadening σ = 0.05 eV so that the peaks of the defect cannot become merged with the band continuum, for the calculations of density of states (DOS).

The band structure and the total DOS of the perfect GaAs and GaAs with defects are shown in Fig. 3 -5 . From Fig. 3, we can see that the perfect GaAs has a direct band gap of 1.5eV at the Г point, which is very close to the experimental result.

Figure 4 shows the band structure of GaAs with the double Ga vacancies. Compared with the perfect GaAs, three defect bands appear upon the top of valence band (the red lines), corresponding to the new peak in right figure of the total DOS, and the Fermi level moves into the defect bands. Relative to its band gap, which has been increased to 1.76 eV at the Г point, the defect bands are near the top of valence band. Thus the double Ga vacancies are p-type doping, acting as shallow acceptors, and provide lots of free holes at the top of valence band, and promote the conductivity of the system. However, no donor defect level emerges in the band gap. Therefore, we can conclude that EL2 defect level is not formatted in GaAs with double Ga vacancies.

In Fig. 5, it can be seen that besides three acceptor defect levels, several donor defect levels emerge below the bottom of conduction band (the red lines). The lowest donor defect level is far away from the bottom of conduction band, locating at 0.85 eV below conduction band, which can be designated as a deep donor level. Therefore, we can conclude that a deep donor defect level is formed by the ternary complex As_{Ga}V_{As}V_{Ga} in GaAs, which is very close to the experimental result of the EL2 defect level.

Further evidence for the electronic structure is obtained from the local density of states (LDOS) shown in Fig. 6
and Fig. 7
. Figure 6(a) shows the total DOS of GaAs with double Ga vacancies (V_{Ga})_{2}, Fig. 6(b) illustrates the LDOS of the nearest As atoms (green atoms in Fig. 1) to the Ga vacancies, and Fig. 6(c) exhibits the LDOS of the special nearest As atom at the middle of the two V_{Ga} (red atom in Fig. 1). We can see that new peaks (red lines) in the gap appear at the same energy position upon the valence band, especially the special red As atom produces more higher peaks. It means that acceptor-like states are present in the gap. From Fig. 1, every green As atom forms bonds with three adjacent Ga atoms, and the red As atom only forms bonds with two adjacent Ga atoms. Therefore, every green As atom has one dangling bond, and the red As atom has two dangling bonds. Clearly the acceptor defect states arise from these V_{Ga}’s nearest-neighbor As dangling bonds.

Figure 7(a) shows the total DOS of GaAs with ternary complex defects (As_{Ga}V_{As}V_{Ga}), Fig. 7(b) illustrates the LDOS of the arsenic antisite As_{Ga} (red atom in Fig. 2), Fig. 7(c) is the LDOS of the nearest Ga atom of V_{As} (orange atoms in Fig. 2), and Fig. 7(d) depicts the LDOS of the nearest As atom of V_{Ga} (green atoms in Fig. 2). The new peaks (red lines) in the gap appear at the same energy position below the bottom of conduction band both in Fig. 7(b) and (c), which suggests that donor-like states are present in the gap. In Fig. 7(d), new peaks (red lines) in the gap appear upon the valence band, which means that acceptor-like states are also present in the gap. From Fig. 2, every green As atom forms bonds with three adjacent Ga atoms, and these V_{Ga}’s nearest-neighbor As dangling bonds induce the acceptor defect states. Every orange Ga atom forms bonds with three adjacent As atoms, and these V_{As}’s nearest-neighbor Ga dangling bonds induce the donor defect states. Specially, the red As_{Ga} atom substitutes for the Ga atom, and forms bonds with three adjacent As atoms, which also induce donor defect states. Therefore, in GaAs with ternary complex defects (As_{Ga}V_{As}V_{Ga}), the acceptor defect states arise from V_{Ga}, and the donor defect states are from V_{As} and As_{Ga}. The lowest donor defect level (position of the green line) corresponds to deep donor defect level, which is very close to the experimental result of the EL2 defect level.

## 3. Molecular dynamic simulation

The evolution process from the double Ga vacancies to the ternary complex As_{Ga}V_{As}V_{Ga} is simulated by *ab initio* molecular dynamic simulation at different temperature, in order to help us to understand the forming micro-mechanism of the EL2 deep-level in GaAs. Atomic positions are updated with MD simulations, where atomic forces are calculated from the Hellmann–Feynman theorem. In order to reduce the concentration of the defects, all the calculations are performed in bigger super-cells of 128 atoms. To lessen the computational cost, calculations are carried out with a grid spacing corresponding to an effective energy cutoff of 300 eV with a single Γ point for the integration in the irreducible Brillouin zone. The equations of motion are integrated using the Verlet algorithm [20, 21] and the integration time step Δ*t* is set to be 3 femtosecond. The structure is relaxed via a special Davidson block iteration scheme and an analytical expression for Hellmann-Feynman forces to achieve a total energy and force convergence at the level of 0.0001eV and 0.001eV/Å, respectively. Simulations are carried out using the NVT ensemble with the temperature controlled with a Nose-Hoover thermostat [22, 23]. The simulation temperature is chosen to be 300K, 773K, and 1173K.

The initial configuration of the double Ga vacancies (V_{Ga})_{2} is prepared by taking two adjacent Ga atoms from a perfect super-cell of 128 atoms, as shown in Fig. 8(a)
. Figure 8(b) shows the simulation results for 2000 steps at 300K, which demonstrates that the special nearest As atom in the middle of two V_{Ga} has migrated to one of the Ga vacancies to be an arsenic antisite As_{Ga}, leaving an As vacancy V_{As} at its own proper position. Therefore, a ternary complex defect (As_{Ga}V_{As}V_{Ga}) is formed. In order to verify the stability of the ternary complex defect (As_{Ga}V_{As}V_{Ga}) in GaAs crystal, we carry out the simulations at higher temperature. The thermalized results for 2000 steps at 773K and 1173K are shown in Fig. 8(c) and (d), respectively. We can see that the ternary complex defect (As_{Ga}V_{As}V_{Ga}) exists in both cases.

Form Fig. 8(a) and Fig. 1, we can see that the special nearest As atom (red atom) in the middle of two V_{Ga} only forms bonds with two adjacent Ga atoms, which is highly unstable at this position. It can escape from its proper position to one of the two Ga vacancies with a minor perturbation, and form bonds with three adjacent As atoms, as shown in Fig. 8(b) and Fig. 2. The red As atom is more stable when the ternary complex defect (As_{Ga}V_{As}V_{Ga}) is formed. The MD results of Fig. 8(c) and (d) verify it further. Therefore, it can be concluded that the evolution process of the double Ga vacancies to the ternary complex As_{Ga}V_{As}V_{Ga} (EL2 defect) is a necessary result.

## 4. Conclusion

In conclusion, first-principles investigations based on the HSE hybrid functional have been carried out for the perfect GaAs crystal, the double Ga vacancies (V_{Ga})_{2}, and the ternary complex defect (As_{Ga}V_{As}V_{Ga}). Our results can be summarized as follows:

- (i) The perfect GaAs has a direct band gap of 1.5eV at the Г point, which is very close to the experimental result.
- (ii) For the GaAs with double Ga vacancies, three shallow acceptor defect bands arise from the V
_{Ga}’s nearest-neighbor As dangling bonds. However, no donor defect level emerges in the band gap, from which we can conclude that EL2 defect level is not formatted in GaAs with double Ga vacancies. - (iii) For GaAs with the ternary complex defect (As
_{Ga}V_{As}V_{Ga}), besides three acceptor defect levels induced by the V_{Ga}’s nearest-neighbor As dangling bonds, the As antisite As_{Ga}and the V_{As}’s nearest-neighbor Ga dangling bonds can induce several donor defect states. The lowest donor defect level locates at 0.85 eV below the bottom of conduction band.

The evolution process from the double Ga vacancies (V_{Ga})_{2} to the ternary complex As_{Ga}V_{As}V_{Ga} has been simulated by *ab initio* MD simulation at different temperature. The MD simulation results demonstrate that the ternary complex defect (As_{Ga}V_{As}V_{Ga}) can be converted from the double Ga vacancies (V_{Ga})_{2} at room temperature, and it can exist stably at higher temperature.

In short, it can be concluded from our calculation that the EL2 deep donor defect level is related to the ternary complex defect (As_{Ga}V_{As}V_{Ga}), which can be converted from the double Ga vacancies (V_{Ga})_{2}. Therefore, by changing the growth conditions of GaAs crystal to control the concentration of the double Ga vacancies (V_{Ga})_{2}, we could control the amount of EL2 defects in GaAs saturable absorber to design the non-linear loss, linear loss, recovery time and the modulation depth.

## Acknowledements

This work was partially supported by the National Science Foundation of China (60876056, 21173134), the founding of the National Municipal Science and Technology Project (No. 2008ZX05011-002), the China Postdoctoral Science Foundation funded project (20090461210), and the Postdoctoral Special Innovation Foundation of Shandong Province (200903067).

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