Investigation Of Oxygen Defects In Wurtzite Inn By Using Density Functional Theory

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Investigation of oxygen defects in wurtzite InN by using density

functional theory

Y. Hattori

a

, J.F.D. Chubaci

a

, M. Matsuoka

a

, J.A. Freitas Jr.

b

, A. Ferreira da Silva

c,n a

Instituto de Física, Universidade de São Paulo, SP 05315-970, Brazil b_{Naval Research Laboratory, Washington DC 20375, USA}

c

Instituto de Física, Universidade Federal da Bahia, Salvador, BA 40210-340, Brazil

a r t i c l e i n f o

Communicated by T. Paskova Available online 31 August 2016 Keywords: Indium nitride Oxygen defects Semiconductors Electronic states Computer simulation Growth models

a b s t r a c t

Density Functional Theory based on ab initio calculations was employed to investigate single and complex defects of oxygen in indium nitride and their inﬂuence on the optical properties. Different oxygen contents (x¼1.38%, 4.16%, 5.55% and 11.11%) were considered in our study by using PBEsol-GGA and TB-mBJ for the treatment of exchange-correlation energy and potential. It was found that oxygen is energetically favorable to exist mainly as singly charged isolated defect. The results using TB-mBJ ap-proximation predicts a narrowing of the VBM (valence band maximum) and CBM (conduction band minimum) as oxygen content increases. Nevertheless, the larger contribution of the Moss-Burstein effect leads to an effective band-gap increase, yielding absorption edge values larger than that of the intrinsic bulk indium nitride.

1. Introduction

InN fundamental parameters values were uncertain for dec-ades. The lack of a suitable substrate material and optimization of growth method resulted on films with high concentration of native defects and poor properties. A band-gap of about 1.9 eV, reported on 1986 [1,2], much larger than the value of about 0.7 eV reported in 2002 [3], illustrate the lack of reliable ex-perimental data. The large difference between these two values is attributed to the lower quality of the InN films deposited by sputtering technique, in the former experiment, as compared with that of thefilms deposited by molecular beam epitaxy in the latter experiment. Thereafter, a number of reports using InNfilms produced by a cleaner deposition method confirmed the smaller InN band gap. More recently, was reported an experimental and theoretical studies of InNfilms produced by modified Ion Beam Assisted Deposition (IBAD) method on GaN/Sapphire substrates verifying an energy gap around 0.8 eV[4]. One of the possible explanation of large the discrepancy in the band-gap energy of InN between previous and more recent data, is the Moss-Burstein effect due to residual large carrier concentrations. Materials like InSb, GaAs are suitable to observe the Moss-Burstein shift be-cause the donor bind energy and the density of states function are quite low due to the low value of the electron effective mass

[5]. Indium nitride also has a strong non-parabolicity of the

conduction band, as indicated in recent works[6,7]reporting an effective mass ofm*0¼0.055 to 0.085 m0. Not intentionally doped

InN has often been found to have very high electron densities, and the n-type conductivity has been attributed to nitrogen va-cancy or to the nitrogen antisite[8,9]. However, Stampﬂ et al.[10]

examined the oxygen impurities and native defects using LDA approximation in InN and they reported that substitutional oxygen ONand silicon SiInact as shallow donors, with formation

energies signiﬁcantly lower than VN. Indeed, oxygen and silicon

are common unintentional impurities in nitrides, and are shallow donors in GaN[11,12]. Although there is not a clear experimental evidence about the role of oxygen in InN, its easy incorporation even in MBE and MOVPE techniques is commonly accepted. In the light of the evidences mentioned above, DFT studies were undertaken to examine the in_{ﬂuence of oxygen in indium nitride} by using PBEsol-GGAþTB-mBJ for exchange and correlation po-tential. In order to achieve a more realistic simulation of the presence of oxygen, weﬁrst investigated the atomic distribution through calculation of the formation energy and the binding energy of the defect. Subsequently, optical properties were ex-tracted for different contents of oxygen. First principles calcula-tion showed that oxygen act as donors and due to the Moss-Burstein effect there is an apparent widening of the band-gap, but interactions between impurity ions and electrons in the conduction band also causes a narrowing between VBM (valence band maximum) and CBM (conduction band minimum). Contents lists available atScienceDirect

journal homepage:www.elsevier.com/locate/jcrysgro

Journal of Crystal Growth

http://dx.doi.org/10.1016/j.jcrysgro.2016.08.058

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2. Theoretical and computational approaches

2.1. Density functional theory calculations

First principles calculations were performed using an aug-mented plane waveþlocal orbitals (APWþlo) approach based on Density Functional Theory (DFT) as incorporated in the WIEN2k code[13]. The bulk structure considered in our work for the in-dium nitride is the wurtzite, which is the hexagonal stable phase. In the computational work presented, In ( p d4 46 105 5s p2 1_{) and N}

(2 2s p2 3) are treated as valence states and local orbitals were used for the semicore states. Since it is well known that GGA potentials underestimate band-gaps[14], the TB-mBJ potential was applied, which usually predicts band-gaps with higher accuracy[15]. It is important to emphasize that mBJ parameters were calibrated such that its potential reproduces at best the experimental band-gaps. The energy was not used at all in this procedure[16], so only the results obtained with PBEsol-GGA were employed to calculate total energies. The heat of formation ΔHfcalculated by PBEsol-GGA

is–0.96 eV and is in good agreement with reported experimental values which range from–0.22 to –1.49 eV[17]. The cohesive en-ergy is calculated to be–10.24 eV and the experimental value is – 7.97 eV [18]. Thus, the PBEsol-GGA approach overestimate the cohesive energy. After the adequate choice of k-mesh and plane wave basis set, the value of the fundamental band-gap of 0.89 eV for the bulk structure was obtained. The structure was optimized through a polynomialﬁtting of a sample of two dimensional var-iation of cell parameters, which for each point an atomic position optimization was done until forces on the atoms fell below 1.0 mRy/a.u. The lattice parameters obtained were a_{¼3.529 Å and} c¼5.751 Å, in reasonable agreement with experimental results

[19]. Thenceforth, a 3 3 2 supercell corresponding to 72 atoms (36 N and 36 In) was created. The atomic spheres used for In, N, and O were 2.20, 1.55 and 1.60 a.u. respectively. The wave func-tions in the atomic spheres were expanded as spherical harmonics up to angular momentum l¼10. In the interstitial region between the atomic spheres, a plane wave expansion was used,ﬁxing the parameter RMT.Kmax¼6, which is the product of the smallest

mufﬁn tin radius and the largest plane wave Kmax. For calculation

of formation energy of oxygen, an especial attention was taken to ensure the use of same atomic spheres radius, basis set size and k-mesh sampling.

The value of the absorption coefﬁcient was obtained through dielectric function in the long wave length q= | ′ −k k| =0 limit from the equations below:

∑

ω π Ω ω δ ω ϵ ( = ) = | < | | ′ > | × ( − ) ( ′− − ) ( ) ′ ′ = e m n n f f E E q 0, 4 k p k 1 ₁ n n n bfk n n n k k k k k 2 2 2 2 2 2 , , , 2 , , , ,

The real part ϵ (1q=0,ω)of the dielectric function can be ob-tained from the Kramers-Kronig relation[20]:

∫

ω π ω ω ω ω ω ω ϵ ( = ) = + ′ϵ ( ′) ′ − + ′ + ( ) −∞ ∞ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ d q 0, 1 1 2 1 1 2 1 2

In the above equation

Ω

is the crystal volume, f_k,_nis the Fermi distribution and |k,n>correspond to the wave function of n-th eigenvalue, Ek,n. The expression of the absorption coefﬁcientα ω( )

is then given by:

α ω( ) = ω − (ϵ( )) + |ϵ( )|ω ω ( ) ⎛ ⎝ ⎜ ⎞_⎠⎟ c Re 2 2 3 1/2

For the calculation of optical band-gap, the absorption coef ﬁ-cient

α

is written as function of energy using the equation:

α ν ν

( h ) =2 A h( −Eg) _{( )}₄

The optical band-gap is obtained then by extrapolating linearly to zero the linear part of the absorption spectrum.

2.2. Formation energy of defects

Calculations of formation energy of defects Δ (E D qf , ) were performed using the Zhang and Northrup formalism[21]:

(

)

∑

(

)

= − + Δ

( )

− μ + + ϵ + ϵ + Δ ( ) E D q E E E q n q V , 5 f D q host el i i i VBM F ,

where ED q, and Ehost are the total energy of the supercells

containing the defect D with charge state q and without defect, respectively. The energy of the added ni¼ þ1 or removed atom

ni¼–1 is referred to the chemical potential of reservoirs for the

different elements

μ

i. For electrons, the chemical potential

de-pends on the Fermi energy

ϵ

F, and the zero is conventionally set as

the valence band maximum of the bulk material

ϵ

VBM. There are

other two terms ΔE qel( )and ΔV to handle spurious image inter-actions in a supercell with charge defects. The monopole Made-lung term, as ﬁrst proposed by Leslie and Gillian [22] were adopted to calculate the former, while the latter was calculated through the total average potential of the defective cell onto the reference average potential of the bulk cell [23]. This procedure was employed to avoid any double counting. The chemical po-tentials depend on the experimental growth conditions. Therefore, they should be considered as variable. However, it is possible to placeﬁrm bounds on the chemical potentials assuming the equi-librium μIn+μN=μInN. For N-rich conditions we have μN = μ

max N

1 2 2,

while for In-rich conditions we have μN = μ min

N

1

2 2þΔHf[InN], where

the experimental value of 0.28 eV was adopted for the heat of formation [24,25]. For the investigation of the incorporation of

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 Exp. bandgap 0.7 eV Exp. bandgap 0.7 eV Exp. bandgap 0.7 eV Ef (eV) εF (eV) E_f of O_N+ (N-rich condition) Ef of ON0 (N-rich condition)

Fig. 1. Formation energy vs Fermi level of isolated O using N-rich condition.

Table 1

Supercell energy for each distribution type.

# Oxygen Uniform (Ry) Cluster1 (Ry) Cluster2 (Ry)

1 427371.752 2 427412.686 427412.678 3 427445.422 427445.419 4 427494.625 427494.650 427494.632 8 427658.222 427658.322 427658.309 12 427821.833 427822.010 427821.951

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substitutional oxygen ONin the nitrogen site it was used μO= μO

1 2 2,

which correspond to the environment in O-rich condition. The calculation of nitrogen and oxygen molecule were also performed using spin-polarized DFT approach with the WIEN2k code. An important concept when investigating complex formation is the

binding energy. It measures the thermodynamic stability of a complex against dissociation into its constituents, which is deﬁned as:

∑

[( … ) ] = [( … ) ] − [ ] ( ) = E X X X E X X X E X 6 b N q f N q k f k q 1 2 1 2 1 k

Fig. 2. Distribution types for a supercell (dashed points represent its boundary) of 72 atoms and 4 (a–c), 8 (d,e) and 12 (f,g) substitutional oxygen. Some atoms were omitted for a easier understanding. (a) Cluster1-type (4 Oxygens). (b) Cluster2-type (4 Oxygens). (c) Uniform type (8 Oxygens). (d) Cluster1-type (8 Oxygens). (e) Cluster2-type (8 Oxygens). (f) Cluster1-type (12 Oxygens). (g) Cluster2-type (12 Oxygens).

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where E X Xf[( 1 2…XN) ]q is the formation energy of the defect complex(X X1 2…XN)in the charge stateq=q1+q2+ ‥qNandE Xf[ k ] q_k

is the formation energy of defect Xkin charge state qk. We adopted

the convention of positive binding energy indicating unstable complexes and negative values indicating stable complexes.

3. Results

3.1. Single oxygen defect: ON

Oxygen has only one more atomic number than nitrogen and is much smaller than indium, so it is expected that acts as a donor when substituting N in InN. Stampﬂ et al.[10]examined the oxygen impurities and native defects in InN using LDA approximation. They reported that oxygen act as donors, with formation energies sig-niﬁcantly lower than VN. We also computed the formation energies

of oxygen atom at the nitrogen site in the neutral and 1þ charge state (Fig. 1). The results show that ONis indeed a single donor in

InN but the transition energy between 1_{þ and 0 charge states are} not de_{ﬁnitive because of approximations involving GGA methods.} The formation energy obtained for ONin InN is a little lower than

that in GaN as reported in another work[26].

3.2. Complex defects

In order to obtain the optical properties of InON compound, we ﬁrst determined what type of distribution for a given small oxygen content has more tendency to form. Thus, different atomic ar-rangements have been investigated for a given oxygen fraction content, by either distributing the impurities as uniformly as possible over the supercell or by clustering the atoms together in a small part of the supercell of 72 atoms. The total energy obtained for each case is listed in Table 1. It was veri_{ﬁed that the lattice} parameter optimization in a relaxed supercell containing 25% of oxygen only decreases the energy in about 0.16 eV. Therefore, lattice parameter optimization was not performed in the struc-tures considered.

It is important to emphasize that in this ﬁrst stage all im-purities were treated in the neutral charge state. For each calcu-lation, the positions of atoms were optimized until forces fell be-low 2.0 mRy/a.u. Up to x¼4.16% (3 oxygen) the lowest energy obtained was the one with uniform distribution, while for x¼5.55% (4 oxygen) the lowest energy value was for the Cluster1-type (Fig. 2a). In the latter case, four atoms tend to surround the indium atom, substituting all 4 nitrogen atoms. For this number of impurities, a situation in which pairs of oxygen were distributed uniformly (Fig. 2b) was also considered, which we will be denoted as Cluster2-type. In the supercell with x¼11.11% (8 oxygen), a

Table 2

Binding energies for each charge state of a 4 oxygen atoms case (Cluster1-type).

Charge state Binding energy per atom (eV)

1þ –

2þ 0.296

3þ 0.558

4þ 0.727

Table 3

Values of optical Egand ΔECBM VBM− extracted fromFig. 3a and b, respectively.

Oxygen (%) Optical Eg(eV) ΔECBM VBM− (eV)

1.38 1.501 0.909 4.16 1.321 0.795 5.55 1.262 0.733 11.1 1.273 0.615 0 5 10 15 20 25 30 35 40 0.6 0.8 1 1.2 1.4 1.6 (α hν ) 2 (a.u.) Energy (eV) InN Bulk 1.38% Oxygen 4.16% Oxygen 5.55% Oxygen 11.1% Oxygen

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0 0.5 1 1.5 2 0 0.05 0.1 0.15 0.2 0.25 0.3 E_g1<E_g2<E_g3 Energy (eV)

Wave vector( 2 π /a.u. ) InN Bulk 1.38% Oxygen 5.55% Oxygen 11.1% Oxygen

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Fig. 3. Narrowing of the band-gap as oxygen content increases. (a) Determination of optical band-gap according to Eq.(4)of InON with different oxygen content using PBEsol-GGAþTB-mBJ method. (b) VBM and CBM of InON with different oxygen content using PBEsol-GGAþTB-mBJ method.

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

1e+19 1e+20 1e+21

Energy (eV)

Electron Concentration (cm−3) Calculated

Yamamoto et al. Walukiewicz et al.

Fig. 4. Theoretical curve of optical band-gap as a function of the carrier con-centration calculated using only PBEsol-GGA method.

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combination of Cluster1- and Cluter2-type was considered, i.e.: the 5.55% oxygen case, where four oxygen atoms closely surround the indium atom (Cluster2-type) (Fig. 2e) and four oxygen atoms were placed further away (Cluster1-type) (Fig. 2d). Finally, a su-percell with x¼16.67% (12 oxygen) was created which they were distributed in three ways: uniform as possible, four atoms clus-tered where each cluster was uniformly distributed (Cluster1-type) (Fig. 2f), and all oxygen atoms clustered in one region (Cluster2-type) (Fig. 2g). Among the last three cases the Cluster1-type is the most likely to form. In short, a transition of preferable distributions from uniform type to a uniformly distributed clusters up to 12 oxygen atoms in the 72 atoms supercell is observed. In face of this results, the study was focused on supercell with four oxygen atoms in the cluster1-type distribution considering the 1þ,2þ,3þ and 4þ positive charge states, and the binding energy was calculated using Eq.(6).Table 2show a list of the obtained binding energies. All considered cases are unstable. In addition, the instability increases as charge state increases, indicating that is energetically favorable for oxygen atoms to exist mainly as singly charged isolated defects. This result agrees with reported experi-mental works[27,28]where no Indium Oxide was observed de-spite high oxygen concentration, as would be expected if clustered distributions were favorable. The calculations of binding energies for other cases were beyond the focus of this work, thus they were not performed (Table 3).

3.3. Optical properties

For the investigation of the optical properties of InON we per-formed thefirst principles calculations of the supercell with 1.38%, 4.16%, 5.55% and 11.11% of oxygen distributed uniformly in space. As it can be seen inFig. 3a the absorption edge decrease as the oxygen content increases, although it is still higher than in the bulk struc-ture. This behavior is due to the narrowing between VBM and CBM (Fig. 3b). Thus, band-gap variations arise from competition between Moss-Burstein effect and the interactions between electrons in the conduction band with the impurities. The same type of calculation was performed using only PBEGGA functional and this behavior of reduction of the optical band-gap was also noticed. The observed trend does not agree with Yoshimoto et al.[29]work where they reported a direct relation between the widening of the optical band-gap with the increasing oxygen content. Yoshimoto's results were contradicted by Wintrebert-Fouquet et al.[30]report. In the latter work, thefilms exhibit single crystalline structure, while in the former thefilms are polycrystalline with low carrier mobility of 10 cm

V s

2

, and no detail about the grain size was provided. Therefore, quantum size effects can't be discarded as is very common in polycrystalline structures and it can strongly affect the absorption spectra. Indeed, Lan et al.[31]reported the observation of 1.9 and 0.77 eV emission bands for InN nanorods of 30–50 nm and higher than 50 nm in diameters, respectively. Chao et al.[32]also obtained a similar result, where the blue shift associated with a decrease in the size of the nanorods from 40 to 5 nm was observed. Un-fortunately, the comparison between experimental results is not possible because there are a hand full of factors that can strongly inﬂuence the optical properties of InN, e.g., quantum size effects, stoichiometry, other sources of impurities, as was widely debated by Butcher et al.[33].

With the purpose to investigate further the inﬂuence of carrier concentration on InN, the variation of absorption edge was cal-culated for different electron concentration using PBEsol-GGA functional (Fig. 4). For each case, the optical band-gap was ob-tained through the ﬁtting of a squared absorption in the linear region (see Eq.(4)). A comparison between experimental results using MBE and MOCVD technique is also displayed. Due to the

underestimation of GGA methods, the calculated curve was dis-placed by 0.7 eV. The graphic shows that Sugita et al.[34]results are in good agreement with our theoretical curve, while Walu-kiewicz et al.[35]results shows a notable discrepancy for higher concentrations. This might be due to the higher electron mobility of 730cm

V s

2

reported by the former author, while the latter obtained 615cm

V s

2

, suggesting higher concentration of compensating defects.

4. Summary

In the present work, we have applied TB-mBJ method to in-vestigate the inﬂuence of oxygen in the indium nitride. The cal-culation of the defect formation energy showed that oxygen can be easily incorporated in the semiconductor and will act as a donor. The results obtained by calculating the binding energy indicates that is energetically favorable for oxygen atoms to exist mainly as singly charged isolated defect. There is a signiﬁcant widening shift in the value of the optical band-gap with the oxygen incorpora-tion, but our results using TB-mBJ predicts a narrowing of the band-gap which compensates the Moss-Burstein effect. Therefore, the latter is more effective than the interactions of the conduction electrons with the ionized impurities. Our calculations of the bulk indium nitride optical band-gap as a function of the electron concentration present good agreement compared to the results of high quality (high mobility) samples.

Acknowledgements

The authors acknowledge theﬁnancial support of the Brazilian agencies FAPESB/PRONEX/ PNX0007/2011, CNPq /303304/2010-3/ PQ and FAPESP. The work was part of an ONR/IF-USP/ NICOP/IF-UFBa project.

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