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Structural, electronic and thermal properties of AlxGa1xAs ternary alloys: Insights from DFT study O. Nemiri a, b, F. Oumelaz a, b, A. Boumaza b, S. Ghemid b, H. Meradji b, *, W.K. Ahmed c, R. Khenata d, Xiaotian Wang e, ** a

ENSET High School of Technological Teaching of Skikda, Algeria Laboratoire LPR, Facult e des Sciences, Universit e Badji Mokhtar, Annaba, Algeria Mechanical Department, College of Engineering, UAE University, Al Ain, United Arab Emirates d Laboratoire de Physique Quantique et de Mod elisation Math ematique, Universit e de Mascara, 29000, Algeria e School of Physical Science and Technology, Southwest University, Chongqing, 400715, People's Republic of China b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 5 May 2019 Received in revised form 18 July 2019 Accepted 19 July 2019 Available online 21 July 2019

In this research paper, we studied the structural, electronic and thermal properties of the zinc blende ternary alloys (AlxGa1xAs) by the use of ﬁrst-principles calculations based on FP-LAPW method (Full Potential Linear Augmented Plane Wave) within DFT (Density Functional Theory). Basically, the impact dependence of the lattice constants, band gaps, bulk moduli, heat capacities, Debye temperatures and mixing entropies on the composition x were investigated for different values of x (x ¼ 0, 0.25, 0.5, 0.75, and 1). The computed ground state properties for the parent binary compounds are in reasonable agreement with the available experimental and theoretical results. It is shown that the lattice constant demonstrated a marginal deviation for AlxGa1xAs alloy from Vegard's law. It was observed for the studied alloy that signiﬁcant deviation of the bulk modulus from LCD (Linear Concentration Dependence). Moreover, it was found that the variation of the energy band gap as function of composition is linear via the mBJ approximation. The thermal parameters of these alloys were investigated by means of the quasi-harmonic Debye model. © 2019 Elsevier Inc. All rights reserved.

Keywords: FP-LAPW DFT AlxGa1xAs Structural properties Gap bowing Thermal properties Debye model

1. Introduction Due to the attractive physical properties, the optoelectronic materials have been studied through many research works. It is well known that IIIeV ternary and quaternary alloys have a wide range of the technological applications, especially in optical detectors, optoelectronic devices, solar cells, and semiconductor lasers. Therefore a great effort has been paid to these alloys recently. The incorporation of aluminum into the standard IIIeV compounds may open pathways for band gap engineering in IIIeV alloys. The structural, electronic and optical properties of aluminum compounds have been investigated by different methods in various experimental and theoretical studies [1e4], and have been reported that the full potential FP-LAPW calculations have been used

* Corresponding author. ** Corresponding author. E-mail addresses: [email protected] (H. Meradji), [email protected] (X. Wang). https://doi.org/10.1016/j.jmgm.2019.07.011 1093-3263/© 2019 Elsevier Inc. All rights reserved.

intensively to estimate the fundamental physical properties of the mentioned compounds. For example, Khanin and Kulkova [5] and Reshak and co-authors [6] used FP-LAPW method with the local density approximation (LDA) to calculate the electronic and optical properties, respectively. Recently, Briki et al. [7] studied the relativistic effects on the structural and transport properties of IIIeV compounds utilizing LDA and PBE-GGA for the exchangeecorrelation energy. Annane and co-workers [8] investigated the structural, electronic and optical properties of AlAs and AlP compounds and their ordered AlAs1-xPx alloy by using the FP-LAPW method with local orbitals. New horizons have been opened recently in condensed matter physics via the ﬁrst-principles theoretical calculations. Computing the electronic and structural properties of solids become possible that allowed researchers to elucidate and predict properties which are experimentally challenging to measure. First-principles predictions in research which is based on the DFT (Density Functional Theory) [9,10] have become an important branch in materials science. The FP-LAPW or pseudo-potential methods [11] were used to

O. Nemiri et al. / Journal of Molecular Graphics and Modelling 92 (2019) 140e146

carry out the majority of these theoretical investigations. Moreover, generalized gradient approximation (GGA) [12] or the DFT within the local density approximation (LDA) [13] were used for these theoretical studies. These approximations yield good results for ground state properties, but on the other hand, for the band gap energy values, they were underestimated. Engel and Vosko (GGAEV) [14] proposed another form of GGA, was explained successfully the electronic properties. Wu and Cohen [15] suggested a new form of the GGA to improve the results of the structural parameters. In order to ﬁgure out the band gap of the materials under study that is closer to the experimental one, the mBJ (modiﬁed Becke-Johnson) exchange potential approximation [16,17] has been adopted. In this work, we have presented the structural and electronic properties of the ternary alloys (AlxGa1xAs) through adopting the full potential LAPW method (Linearized Augmented Plane Wave) within the DFT for studying the composition dependence of the band gap, bulk modulus and lattice constant. Thermal properties were also investigated by means of the quasi-harmonic Debye model. No experimental or theoretical studies have been reported so far to explore thermal properties, i.e., the entropy (S), Debye temperature (qD), heat capacity (CV), as a function of temperatures. For this important material, our aims are to present and add reference data set to the existing theoretical and experimental studies. The rest of the paper has been divided in three sections. In Section 2, a description of the method of calculation and approximations used in the present investigation are given. In Section 3, for the AlxGa1xAs, the results and discussion for the most substantial ﬁndings estimated for the electronic, thermal and structural properties are shown. A summary of the main conclusions is given in Section 4. 2. Method of calculations FP-LAPW (full potential linearized augmented plane wave) method [18] was adopted in this work, as contained in WIEN2K computer software [19]. In this software, the crystal structures were optimized under the circumstances that the total energy is minimum with respect to atomic coordinates for a given crystal symmetry. The exchange and correlation potential was treated by the generalized-gradient approximation (GGA) using the Wu and Cohen (WC) parameterization [15] which is an improved form of ordinary GGA due to Perdew-Burke-Ernzerhof (PBE-GGA) [12]. Furthermore, the Tran-Blaha modiﬁed Becke-Johnson (TB-mBJ) potential [16,17] was also applied to obtain reliable results for the electronic properties of the ordered AlxGa1xAs alloy and their parent compounds. The quasi-harmonic Debye model [20] was used to study the thermal properties. For a solid described by an energy-volume (E-V) relationship in the static approximations, the Gibbs program can be used to perform different tasks, e.g., to obtain the Gibbs free energy G (V, P, T), and to calculate the Debye temperature. Standard relations of thermodynamic were used to derive the other macroscopic properties correlated to pressure and temperature. The calculation details can also be found in Refs. [21e24]. For the charge density, potential and the wave functions were expended differently using FP-LAPW method over the unit cell divided into two regions. Spherical harmonics expansion is adopted within the non-overlapping spheres of radius RMT about each atom, whereas the plane waves were chosen for the interstitial region. The parameter selected in the calculations that estimate the size of the secular matrix, is RMT * Kmax ¼ 8, in which the RMT represents the Mufﬁn-Tin sphere radii and Kmax represents the cut-off wave vector within the ﬁrst Brillioun zone (BZ). The mufﬁn-tin spheres that includes valence wave functions are prolonged up to lmax ¼ 10. Gmax ¼ 14 (Ryd)1/2 was taken for the Fourier component of the

141

charge density. The radii of the Mufﬁn-tin (i.e., RMT) spheres were proposed to be 2.17, 2.1, and 2.0 a.u. for Ga, As, and Al atoms, correspondingly. For binary compounds and ternary alloys, a mesh of 72 and 36 special k-points, respectively, were considered in the irreducible wedge of the Brillouin zone (IBZ) for the complete energy prediction. The numbers of k-points and the plane wave cutoff were selected to achieve high degree of total energy convergence. 3. Discussion of the results 3.1. Structural properties Firstly, alloys were modeled over a speciﬁc nominated compositions (x ¼ 0.25, 0.5, and 0.75) with systematic structures deﬁned regarding periodically recurrent supercells. Simple cubic cells with eight atoms were utilized for the studied materials. Secondly, the structural optimization was executed via diminishing the entire energy in regard to the cell parameters as well as to the atomic locations. Finally, calculations have been performed to estimate the total energy with respect to the volume of the unit cell and are approximated by curve ﬁtting to the equation of state (EOS) of Murnaghan [25]. The equilibrium structural properties including the bulk modulus (B) (and its corresponding pressure derivative (B0 )) and the lattice constant (a) were predicted from the EOS. A comparison was performed for the equilibrium lattice constants for each x of the AlxGa1xAs alloy with respect to the previously published theoretical and experimental studies, as displayed in Table 1. It is concluded that the lattice constant for GaAs (as shown in Table 1) calculated here is slightly larger than the experimental [26] and theoretical ﬁndings [28e31]. The minimum value of lattice constant is obtained as 5.668 Å for GaAs compound but the maximum value is obtained as 5.680 Å for AlAs. Hence, from these results we can observe that the lattice constants of AlxGa1xAs will increase as the concentration of Al increases. For AlxGa1xAsx with WC-GGA in zinc blende structure, Fig. 1 presents the calculated lattice constants at various compositions x. The lattice constant demonstrated almost linear behavior in accordance with Vegard's law [33] with a marginal upward bowing parameter equals to 0.00331 Å. This value is higher than that obtained by F. El Haj Hassan [28]which was - 0.008 Å and is lower than the experimental value that equals to þ0.007 Å [34]. This bowing parameter is determined by ﬁtting the calculated lattice constants versus composition to a polynomial function. The physical origin of this deviation must be mainly attributed to the mismatches of the lattice constants of GaAs and AlAs compounds. Also, it should be related to the size of atoms, the ratio R(Al)/R(Ga) ¼ 1.10, where R is Table 1 Calculated lattice parameter (a, in Å) for AlAs, GaAs compounds and their ternary AlxGa1xAs alloys. Alloy

x

Lattice constant a (Å)

Exp.

Other calculations

AlxGa1xAs

0 0.25 0.5 0.75 1

5.668 5.671 5.675 5.678 5.680

5.653a

5.666c, 5.58d, 5.53e, 5.54f

5.66b

5.731g, 5.74h, 5.633i, 5.734i

a b c d e f g h i

Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[26]. [27]. [28]. [29]. [30]. [31]. [8]. [7]. [32].

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O. Nemiri et al. / Journal of Molecular Graphics and Modelling 92 (2019) 140e146

Fig. 1. Composition dependence of the predicted lattice constant of AlxGa1xAs alloy (solid squares) in comparison with Vegard's law (dashed line).

the atomic radius. It is well known, when an atom is replaced by a smaller one, the value of the lattice constant will decrease; whilst substitution with a bigger atom will increase the value of the lattice constant. Table 2 illustrates our theoretical bulk modulus for zinc blende structure. A small deviation is clearly observed between the computed values of the bulk moduli and their corresponding experimental and theoretical values. Besides, it is noticed that for GaAs, the bulk modulus is lower than that of AlAs. That is to say, the increase of x concentration raises the bulk modulus. The variation of the bulk modulus (B) as a function of the alloying concentration x for the AlxGa1xAs compounds are depicted in Fig. 2, and compared to the estimated result by the linear concentration dependence (LCD). We can see that as long as Al concentration increases, the levels of the bulk modulus increase, indicating that contribution of doping Al to GaAs can improve the hardness of this material. An important deviation from LCD is witnessed with upward bowing estimated around - 2.167 GPa. The large bowing value is attributed to the signiﬁcant mismatch between the bulk modulus of the binary compounds AlAs and GaAs where they are equal to 68.782 and 71.464 GPa for GaAs and AlAs, respectively. For the alloys under study, the lattice constants are compared with the bulk modulus, where Figs. 1 and 2 demonstrate that an increase of the former and the later parameters are correlated to each other. It characterizes bonding strength or weakness properties that are attributed to the changing composition. To show the mechanical stability of the structures for different concentrations as an example, the elastic constants of Al0.5Ga0.5As

Fig. 2. Composition dependence of the calculated bulk modulus of AlxGa1xAs alloy (solid squares) compared with the linear composition dependence estimation (dashed line).

have been calculated. A cubic crystal has three independent elastic constants (C11, C12 and C44). They are important in providing valuable information about the mechanical stability of materials. The requirement of mechanical stability in a cubic crystal leads to the following restrictions on the elastic constants [36]:C11 - C12 > 0, C11 > 0, C44 > 0, C11þ 2C12 > 0 and C12< B < C11. The obtained values are C11 ¼129.39 GPa, C12 ¼ 38.91 GPa and C44 ¼ 108.19 GPa, which obey the stability criteria, indicating that the structure is mechanically stable. 3.2. Electronic properties In general, for electrons in a solid, the microscopic behavior is mostly suitable with respect to the electronic band structure. Optimized atomic positions can be employed to estimate the electronic band structures at the equilibrium lattice constants at P ¼ 0 GPa and T ¼ 0 K. The modiﬁed Becke-Johnson (mBJ) approximation has been used to calculate band structures of AlxGa1xAs ternary alloys. Fig. 3(aee) presents the band structures of the binary compounds and corresponding ternary alloys at zero pressure along the principal symmetry directions in the Brillouin zone. The zero energy is selected to match the upper level of valence band. It is obvious from Fig. 3 (a - b) that the CBM (Conduction Band Minimum) is positioned at the G and X points for GaAs and AlAs, respectively. The VBM (Valence Band Maximum) is located at the G point for these two compounds, causing indirect and direct ((G eX) and (G e G)) band gaps for AlAs and GaAs, respectively. On the

Table 2 Calculated bulk modulus (B, in GPa) for AlAs, GaAs compounds and their ternary AlxGa1xAs alloys. Alloy

x

Bulk Modulus B (GPa)

Exp.

Other estimations

AlxGa1xAs

0 0.25 0.5 0.75 1

68.782 70.046 70.469 71.494 71.464

77.00a

74.53c, 75.73d, 77.10e, 69.6f

82a, 78.1b

67.732g, 66.8h

a b c d e f g h

Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[27]. [35]. [29]. [30]. [31]. [28]. [8]. [7].

O. Nemiri et al. / Journal of Molecular Graphics and Modelling 92 (2019) 140e146

143

Fig. 3. Band structure along the symmetry lines of Brillouin zone at the equilibrium lattice constant calculated using the mBJ (GaAs, AlAs, Al0.25Ga0.75As, Al0.5Ga0.5As and Al0.75Ga0.25As).

other hand, at a range of the composition value x considered (i.e., 0.25, 0.50 and 0.75), the AlxGa1-xAs ternary alloys have direct band gaps (G e G). Table 3 shows the estimated values of the band gaps for the investigated composition (x) values (0, 0.25, 0.50, 0.75) that are compared with the previous published data. In the present study, the predicted band gaps based on mBJ approximation demonstrated a remarkable enhancement in comparison with former published theoretical studies and showed a good agreement with the experimental data. Fig. 4 illustrates the band gap variation with respect to composition. Clear evidence shows that Al concentration increases the energy band gap. The lowest value of energy band gap (1.56 eV) was estimated for GaAs, and the largest value of energy band gap (2.161 eV) was obtained for AlAs. The predicted energy band gap values at numerous concentrations are ﬁtted to a 2nd order polynomial equation:

Table 3 Band gaps energy Eg (eV) of AlxGa1xAs ternary alloys at various compositions. Alloy

x

Eg (eV)

Exp.

Other calculations

AlxGa1xAs

0 0.25 0.5 0.75 1

1.560 1.821 1.981 2.049 2.161

1.52a

0.966c, 1.51d, 1.52e, 1.20f

2.170b

2.104g, 1.43h, 1.39i, 1.31j, 1.40j, 2.25j

a b c d e f g h i j

Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref. Ref.

[37]. [26]. [28]. [38]. [39]. [40]. [29]. [7]. [6]. [32].

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O. Nemiri et al. / Journal of Molecular Graphics and Modelling 92 (2019) 140e146

( ) d2 EðVÞ Bs yBðVÞ ¼ V dV 2

(5)

f ðsÞis expressed by:

8 2 31 91 3 > > > < 6 21 þ s 3 2 11 þ s3 2 7 > = 7 6 f ðsÞ ¼ 342 þ 5 > > 31 2s 31 s > > : ; =

=

(6)

=

In Eq. (6), the Poisson ratio s is considered as 0.25 [41]. Consequently, the non-equilibrium Gibbs function G*ðV; P; TÞthat is correlated toðV; P; TÞ can be optimized with respect to volume V, as shown in Eq. (7):

vG*ðV; P; TÞ ¼0 vV P;T

Fig. 4. Band gap energies of AlxGa1xAs versus composition x.

Eg ¼ 1. 572 þ 1. 017 x - 0. 445 x2

(1)

From this equation, the energy band gap values of the parent compounds GaAs and AlAs can be computed.

Thermal equation of state (EOS),VðP; TÞ can be obtained through solving Eq. (7). The heat capacity CV and the thermal expansion coefﬁcienta, the isothermal bulk modulus BT, are extracted from Eqs. (8e11) [21]:

BT ðP; VÞ ¼ V

3.3. Thermal properties Sometimes the experiments are not applicable for a certain values of temperature and pressure, or even being unapproachable to experiment, thermal functions tools offering important solutions to estimate the material characteristics at theses exceptional conditions or in those regions of pressure and temperature where experiments are not available. The thermal properties of AlxGa1xAs alloys were investigated by using the quasi-harmonic Debye model [21]. As a ﬁrst step, a set of total energy calculation versus primitive cell volume (E-V), in the static approximation, was carried out and ﬁtted with the numerical EOS in order to determine its structural parameters at zero temperature and pressure, and then derived the macroscopic properties as function of pressure and temperature from standard thermodynamic relations. In this model the nonequilibrium Gibbs function G*(V; P, T) in the quasi-harmonic Debye model, can be formulated as below:

G*ðV; P; TÞ ¼ EðVÞ þ PV þ AVib ½qðVÞ; T

(2)

whereqðVÞ is the Debye temperature,EðVÞ is the total energy per unit cell volume, PVcorresponds to the constant hydrostatic pressure-volume condition, and AVib is the vibrational term, which can be expressed using the Debye model of the phonon density of states as shown below [22,23]:

0 3 2 1 q qD 9 q AVib ðq; TÞ ¼ nkB T 4 D þ 3 [email protected] e T A D D 5 8T T

(3)

where D qTD represents the Debye integral, nis the number of atoms per formula unit. Basically, for isotropic solid, qD is formulated as shown below [22]:

qD ¼

rﬃﬃﬃﬃﬃ Z h 2 12 i1=3 Bs 6p V n f ðs Þ kB M

(4)

where BS is the adiabatic bulk modulus, approximated by the static compressibility and M is the molecular mass per unit cell [21].

(7)

d2 G* ðV; P; TÞ dV 2

(8) P;T

i h q 3ln 1 eqT S ¼ nk 4D T 3q=T CV ¼ 3 nkB 4Dðq=TÞ q=T1 e

a¼

gC V BT V

(9)

(10)

(11)

where g is the Grüneisen parameter that is expressed by:

dlnqðVÞ dlnV

g¼

(12)

The thermal properties of AlxGa1xAs (x ¼ 0, 0.25, 0.5, 0.75 and 1) are determined in the temperature range from 0 to 800 K, a range where the quasi-harmonic model remains fully valid. The temperature dependence of the lattice constant and bulk modulus of ternary alloys is illustrated in Figs. 5 and 6, respectively. From Fig. 5, it is concluded that when temperature increases, the lattice parameter increases; also we can note that the increase in temperature is associated with the rise of lattice parameter at each concentration. We have presented in Fig. 6 the change of bulk modulus as a function of temperature. Since it can be obtained by the second derivatives of the internal energy with respect to strains, the bulk modulus B is related to interatomic potentials. The anharmonic interactions are reﬂected partially by the temperature dependence of bulk modulus. This is attributed to the bulk modulus would be independent of temperature for a purely harmonic crystal. A clear and remarkable reduction in the bulk modulus with the increasing of temperature can be found in Fig. 6. The Debye temperature (qD) is a valuable thermal parameter since it is inherently associated to the lattice vibrations. In fact, it is a measure of the vibrational response of the material and hence, intimately associated with properties like the speciﬁc heat, thermal expansion, and vibrational entropy. Fig. 7 shows the Debye temperature variation with respect to the concentration x as a function of temperature. Fig. 7 shows that qD is almost constant for the range

O. Nemiri et al. / Journal of Molecular Graphics and Modelling 92 (2019) 140e146

Fig. 5. Variation of the lattice constant (a) with temperature for AlxGa1xAs ternary alloys.

145

Fig. 8. Variation of the heat capacity (CV) as a function of temperature for AlxGa1xAs ternary alloys.

Fig. 9. Variation of the entropy (S) as a function of temperature for AlxGa1xAs ternary alloys. Fig. 6. Variation of the bulk modulus (B) for AlxGa1xAs ternary alloys.

Fig. 7. Variation of the Debye temperature (qD) for AlxGa1xAs ternary alloys.

0e100 K and drops linearly thereafter as temperature increases. Moreover, it is observed as well in Fig. 7, that qD increases with Al concentration at a ﬁxed temperature. This prediction is in accord

with the conclusion that qD is proportional to the bulk modulus and that a high Debye temperature implies hardness of a material. The understanding of the heat capacity of a substance will provide with vital information regarding vibrational characteristics as well as for other thermal properties. Fig. 8 illustrates the estimated heat capacity at constant volume (CV) as a function of temperature for different values of x (0, 0.25, 0.5, 0.75 and 1). It is observed from Fig. 8 that at low temperature, the values of CV increase, at intermediate temperatures this increase becomes gradual and at high temperatures it tends to the DulongePetit limit, which is the same for all solids at elevated temperature levels. This means that CV is proportional to T3, at low temperatures, which is due to anharmonic approximation of the Debye model [42]. However, at temperatures mid-level, the dependence of CV is governed by the vibrational details of the atoms and for a long time that could be only predicated experimentally. At elevated temperature, the CV tends to the Dulong-Petit limit [43]. Entropy (S) describes the dispersal of energy and matter. A measure of disorder of a system is the best deﬁnition of the entropy microscopically. Fig. 9 illustrates the variation of the entropy (S) as a function of temperature. A sharp increase in entropy is observed as the temperature values increase. This is due to the fact that the increasing temperature will lead to the increasing vibrational contribution.

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Besides, there is insigniﬁcant change in entropy as the composition values x changed, whereas for high temperature, as the Al concentration increases, the entropy S decreases. 4. Conclusions The electronic, thermal and structural properties for AlxGa1xAs alloy have been investigated through the ﬁrst principles prediction in the framework of DFT within the WC-GGA and mBJ approximations. The theoretical results show excellent consistency with the experimental results of lattice parameter and bulk modulus; that validate the methodology adopted in the present analysis. The lattice parameter follows Vegard's law for AlxGa1xAs, whereas it is observed that a substantial deviation of the bulk modulus from the LCD that is essentially attributed to the mismatch of the bulk modulus of GaAs and AlAs binary compounds. The use of mBJ scheme to calculate the band gaps leads to a better agreement with experimental data for GaAs and AlAs bulk materials. Finally, for a comprehensive fundamental properties of the alloys analyzed in this study, a detailed investigation is presented of the thermal properties by means of the quasi-harmonic Debye-model. The heat capacity, entropy and the Debye temperature as a function of temperature are estimated systematically within the range 0e800 K. We noticed that the calculated heat capacity Cv of AlxGa1xAs alloys approach the DulongePetit limit, which is typical to all solids at elevated temperatures. The thermal properties investigated in this work would be useful for the anticipated future experimental and theoretical investigations. Acknowledgements The author (R.K.) would like to acknowledge the help of S. H. Naqib from University of Rajshahi- Bangladesh for his careful reading of the paper. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.jmgm.2019.07.011. References [1] P.V. Seredin, A.V. Glotov, E.P. Domashevskay, I.N. Arsentyev, D.A. Vinokurov, I.S. Tarasov, Physica B 405 (2010) 4607. [2] A.R. Jivani, H.J. Trivedi, P.N. Gajjar, A.R. Jani, Pramana - J. Phys. 64 (2005) 153. [3] S. Zh Karaahanov, L.C. Yan Voon, Semiconductors 39 (2) (2005) 161.

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