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Structural and elastic properties of YP1-xAsx alloys from first principles calculations.
Chaouche Yassine1,* and Mourad Souadkia2
1University of Larbi Tébessi, Tebessa, Laboratoire de Physique Appliquée et Théorique, Route de Constantine 12002 Tebessa, Algeria
2Physics Laboratory at Guelma, Faculty of Mathematics, Computing and Material Sciences, University 8 Mai 1945 Guelma, P.O. Box 401 Guelma 24000, Algeria
*E-mail: [email protected]
We have investigated the structural and elastic properties of transition-compound YP1-xAsx alloys in the range from 0 to 1; x=0, 0.25, 0.5, 0.75 and 1 using first principles calculations in the ABINIT code within the generalized gradient approximation (GGA) as the exchange correlation term. With the respect of the concentration (x) of As atom, we computed the variation of the structural parameters, the elastic constant, the optical and acoustic phonon frequencies at the high symmetry points ?, X and L, the static dielectric constant, electronic dielectric constants and the born effective charge by using the virtual crystal approximation (VCA). All the results obtained are fitted by quadratic relations and obtained the corresponding bowing parameters.
Keywords: VCA, YP1-xAsx alloys, elastic properties, Pseudopotentials, dielectric constants.

Today, the calculations of quantum mechanical properties have reached a satisfactory level of difficulty to reproduce satisfactorily experimental results or to study some important properties which we haven’t the experimental data. However, these parameters as structural and elastic properties may be very good analysis to estimate the quality of a theoretical approach. As a matter of fact, they need calculations, step by step, of the hyper surface of the total energy for appropriate lattice deformations and numerical calculation of the second derivatives of the energy as function of the strain compounds 1.

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In the fact, many physical properties such as; band gap, lattice parameters, elastic properties, optical and magnetic properties of compounds can be different scientifically by adding of another material of known properties. The principal reason of this work is to further the understanding of the lattice parameters and elastic properties of the YP1-xAsx alloys which the concentration varied from 0 to 1in the stable phase Rock-salt.

The yttrium pnictides compounds YX (X=N, P, As and Sb), have a high pressure research on both experimental and theoretical methods in particular on the structural phase transition and elastic constants 2–11 because of these compounds are strongly correlated systems 12. They illustrate some attractive properties such as hardness and high melting point; these properties use these compounds in the very large applications in technology as coating and magnetic storage 13. Also, in optoelectronic devices like light detectors and emitting 14.
These materials have two phases, one in the ambient conditions called NaCl, is a stable phase and unstable phase CsCl under pressure 15,16.

There are a lot of studies of yttrium pnictides compounds YX (X=N, P, As and Sb) concerning structural phase stability, electronic properties, dynamical properties, thermodynamic properties and optical properties using theoretical method such as FP-LAPW and pseudopotentials 17-21. Recently, first principles calculations with pseudopotential method were used to obtain the stable and the unstable phase from phonon spectrum and thermodynamic properties yttrium pnictides compounds 15. The part of experimental studies on these compounds is fewer predicted in the literature; the structure and phase transition under pressure of YSb compound using synchrotron and Powder X-ray diffraction 5.
The elastic properties of these materials are less studied; however there is no study, in the limit of our knowledge, on the ternary YP1-xAsx alloys by using VCA. In this article we focus our study in the composition dependence of the lattice parameter a described by Vegard’s law, bulk modulus (B), its pressure derivative B0 and elastic constants (C11, C12, C44).

This article is as follow; description of the detail calculations, the discussion of the obtain results by pseudopotentials method and in the end conclusion are given.

Details of the calculations
In this studies, our calculations is based on the VCA to obtain the structural and elastic properties of YP1-xAsx alloys as function of x concentration by As atom, were done using the ABINIT package 22-25 in the Density functional theory (DFT) within the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) 26 parameterization for the exchange-correlation potential. To describe the interaction between the valence electrons and the nuclei and core electrons we used pseudopotentials of Troullier and Martins type, generated with FHI code (Fritz-Haber-Institut) 27-29 for Y, P and As. To agree a very well convergence, we have doing convergence tests demonstrate that the Brillouin zone sampling by a 4x4x4 Monkhorst Pack 30 mesh of k points and the kinetic energy cutoff are sufficient for the plane wave basis of 80 Hartree.

To calculate the elastic constants, we use the perturbation theory approach of Hamannet al. 31 which based in ABINIT code. In the other hand we obtained the born effective charges, the dielectric constants by using density functional perturbation theory 32–36. The virtual crystal approximation VCA is applied to treat the disordered ternary alloys. The retention of a crystal with primitive periodicity is allows. This is composed of virtual atomic potentials averaging those of the atoms in the parent compounds, so the alloy pseudopotentials are constructed within a first principles VCA scheme. The pseudopotentials of P and As are combined to create the virtual pseudopotentials of the (P1-xAsx) atom. In this work we used, in different composition x, the atomic positions of Y, P and As which it is listed in table1.
VVCAPSx=xVAsPS+(1-x)VPPS……………………………………….. (1)
Where VAsPS and VPPS are the pseudopotenitals of As and P respectively.
Table1. Atomic positions
As concentration (x) Atomic Atomic positions
0.25 Y (0 0 0), (1/2 1/2 0), (1/2 0 1/2), (0 1/2 1/2)
P (1/2 1/2 1/2), (1/2 0 0), (0 1/2 0)
As (0 0 1/2)
0.50 Y (0 0 0), (1/2 1/2 0), (1/2 0 1/2), (0 1/2 1/2)
P (1/2 1/2 1/2), (1/2 0 0)
As (0 1/2 0), (0 0 1/2)
0.75 Y (0 0 0), (1/2 1/2 0), (1/2 0 1/2), (0 1/2 1/2)
P (1/2 1/2 1/2)
As (0 0 1/2), (0 1/2 0), (0 0 1/2)
3. Results
3.1 Structural properties and phase transition
The Yttrium pnictides alloys YP1-xAsx have been investigated in NaCl phase to show the effect of Arsenic composition on the structural properties; lattice parameters, bulk modulus and compared with Vegard law’s 37. Firstly, the equilibrium lattice parameters have been computed using the minimization of the crystal total energy that is a function of different values of the lattice constants. Secondly, by using the Murnaghans equation of states (eos) 38, we have fit the calculated total energy to determine the lattice constant and bulk modulus by varying the arsenic concentrations in the ternary alloy for x = 0, 0.25, 0.50, 0.75 and 1. At normal conditions, the binary compounds crystallize in the B1 phase with experimental lattice parameters a = 5.652 A° 39 and a = 5.786 A° 40 for YP and YAs respectively, which here in this work at x=0 YP is about 5.659 A° and at x=1 YAs is about 5.818 A°, it is clear that the results obtained of equilibrium lattice are in good agreement with other theoretical data using other computation methods and available experimental values as exposed in table 2 for concentration x=0 to 1. However, for YP1-xAsx alloys as function of composition x , we have fitted the lattice constants with a quadratic relation, in general, of AB(1-x)Cx :
aAB(1-x)Cx=xaAC+1-xaAB-bx(1-x) ………………………….. (2)
Where; aAC and aAB are the equilibrium lattice constants of the binary compounds AC (YAs) and AB (YP), respectively. aAB(1-x)Cx (YP(1-x)Asx ) is the alloy lattice constant and the quadratic term b stands for the bowing parameter. The obtained values using the GGA are listed in Table1 for YP1-xAsx alloys with various composition x at x=0, 0.25, 0.50, 0.75 and 1.

An increase of lattice constant with the addition of Arsenic composition is presented in Figure1.

Figure1. Lattice constant versus Arsenic concentration.

The variation of computed lattice constant with Arsenic concentration is approximately linear, and shows a minor deviation from Vegard’s law with a small growing bowing parameter of – 0.01 Å.
The lattice parameter for YP (x = 0) is less than those of YAs (x = 1), while the bulk modulus value for YP is larger than those of YAs, showing an opposite rapport between bulk modulus and unit cell volume.

Figure2. Bulk modulus as function of the Arsenic concentration.

In Figure 2 is shown the fact of bulk modulus as a function of As-composition for YP1-xAsx alloys, and compared to the results predicted by Linear Composition Dependence (LCD).
A considerable deviation from LCD with downward bowing equal to 0.816 GPa forYP1-xAsx, alloys. Another physical reason of the relation of the As concentration and bulk modulus, is might be, the bulk modulus decreases by the increase of the increment of As concentration (x), the alloys become usually more compressible.

In general, the bulk modulus B relates the change in volume of a material with a change in pressure (assuming constant temperature):
? = ?? ??/??, If you decrease (increase) all dimensions of your cell simultaneously (as you did for the E(V) calculation), this corresponds to putting positive (negative) external pressure on your cell. The pressure (at constant temperature) follows from the energy change according to: ? = ? ??/??. Therefore, for the bulk modulus we find:
? = ? ?²? /??²
Table 2 : Structural parameter, lattice parameter a0 in (ºA), Bulk modulus B in (GPa), bulk modulus pressure derivative B’ of YP(1-x)Asx alloys.

Alloys with As concentration (x) parameters This work (VCA) Other. Expt
(x=0) YP a0 5.659 5.659a, 5.673b, 5.683c , 5.6439d 5.652e
B 81.250 79.99a, 86.285c , 87d –
B’ 3.500 3.64a, 3.805c , 3.68d –
(x=0.25) YP0.75As0.25 a0 5.702 – –
B 78.461 – –
B’ 3.851 – –
(x=0.50) YP0.50As0.50 a0 5.742 – –
B 75.760 – –
B’ 3.828 – –
(x=0.75) YP0.25As0.75 a0 5.780 – –
B 73.125 – –
B’ 3.742 – –
(x=1) YAs a0 5.818 5.818a, 5.835 c , 5.7665 d 5.786f
B 70.679 70.012a, 76.198 c , 77 d –
B’ 3.725 3.563a, 3.821 c , 3.89 d –
a Ref. 15,using PP-PW (GGA). b Ref. 16,using PP-PW (GGA). c Ref. 4, using FP-LAPW (GGA). d Ref. 3,using FP-LAPW (GGA). e Ref. 39, at room temperature. f Ref. 40, experimental data.

In the normal conditions the yttrium (Y) compounds YN,YP, YAs and YSb take the NaCl form structure (B1) with Fm3m (#225) space group. These materials, under pressure, change from the sixfold coordinated NaCl phase to the eightfold coordinated CsCl type phase (B2). This structure transition is a first order phase transition. The (B2) structure is in space group symmetry Pm3m (#221). The table3 Present the results of transition pressure of YP(1-x)Asx alloys at different concentrations of As, from x=0 to 1 with other available theoretical data. The results of transition phase plotted in figure 3 and it show that the values of Pt increase with the increasing of the concentration of Arsenic As from 0 to 1. The quadratic relation is
P is the phase transition and b is the bowing parameter, the value of b is ……..

Table 3 : Phase transition pressure Pt of YP(1-x)Asx alloys.

Alloys with As concentration (x) This work (VCA) Other. Expt
(x=0) YP Pt 72.5 72.5 15, 63.83, 55.944 –
(x=0.25) YP0.75As0.25 Pt 72,77 – –
(x=0.50) YP0.50As0.50 Pt 72,97 – –
(x=0.75) YP0.25As0.75 Pt 73,89 – –
(x=1) YAs Pt 75 7515, 58.253, 50.454 –

Figure3. Phase transition as function of the Arsenic concentration.

Elastic properties
The cubic crystal is usually defined via three independent elastic constants. The C11 tensor provides information concerning resistance in the direction of <100>. The C44 expresses resistance against shear deformation along the plane (100) in the 110 direction. The other elastic constant namely C12 have not physical significant, except if we do the arrangement of C12 with C11 and C44 gives other information a propos the elastic behaviour of the compounds. However, the computed of elastic constants provide the mechanical stability conditions for the cubic crystal which satisfy the relations 41,42:
C11- C12 > 0, C11 > 0, C44 > 0 and C11+ 2 C12> 0
In this study, the obtained values of C11 for yttrium alloys YP(1-x)Asx is greater than their C12 and C44 values for each concentration of Arsenic. This high value of C11 is an indication that the compression resistance is more resistant than shear. By comparing the values of elastic constants of yttrium alloys YP(1-x)Asx as a function of concentration (x) of Arsenic; x=0, 0.25, 0.50, 0.75 and 1, a decrease in the C11 and C44 values is observed. Which indicate that YP(1-x)Asx is less resistant to shear deformation with the increase of concentration(x).

The results obtain by the pseudopotentials method with VCA approximation, show that satisfy the mechanical stability criteria.

In the current paper, the calculated elastic coefficient C11, C12 and C44, of YP(1-x)Asx are shown in Table 4 and plotted in Figure.4 with different concentration of As. It can be observed that the elastic constants decrease with As composition (x). Their dependence on As composition(x) is quadratically fitted using the following relation:
CijAB(1-x)Cx=xCijAC+1-xCijAB-bijx(1-x) …………………. (3)
We defined; CijAC (CijYAs) and CijAB (CijYP) as elastic constants of binary materials AC (YAs) and AB (YP) respectively. CijAB(1-x)Cx (CijYP(1-x)Asx ) is the alloy elastic constants and bij is the bowing parameter. These parameters are obtained about b11= 7.457, b12= 0.368 and b44=0. 216 GPa.

Figure 4: Elastic constants to respect the composition (x) of Arsenic.

Table4: Elastic constants in GPa with GGA approximation of Yttrium alloys.
Alloys with As concentration (x) parameters This work (VCA) Other. Expt
(x=0) YP C11 182.4341 204.66 13, 116.043, 20544, 199.445 –
C12 32.853 20.0713, 31.043, 3944, 28.745 –
C44 52.011 40.4613, 86.2843, 2144, 46.345 –
(x=0.25) YP0.75As0.25 C11 181.9102 – –
C12 31.6258 – –
C44 50.3965 – –
(x=0.50) YP0.50As0.50 C11 168.2177 – –
C12 30.5120 – –
C44 47.9940 – –
(x=0.75) YP0.25As0.75 C11 162.8106 – –
C12 29.3761 – –
C44 45.9081 – –
(x=1) YAs C11 157.731 184.2213, (192)44, 177.845 –
C12 28.355 30.7513,233, 4144, 24.445 –
C44 43.869 42.7113, 393, 1944, 3845 –
Dynamic properties
3.3.1. Phonon frequencies
The table 5 listed the phonon frequencies with other theoretical, with other calculation methods, and they agree very well with other data in x=0 and x=1. The results are plotted in figure 5 at ?, X and L points.

Figure 5: phonon frequencies at high symmetry ?, X and L points to respect the composition (x) of Arsenic.

The equation can be used to predict the frequency of any mode of Arsenic concentration x in the studied ternary alloys.
We note that for low (high) As concentration in YP(1-x)Asx, the quadratic fit over estimates (underestimates) the calculated values of longitudinal optical mode frequencies. The obtained values for the bowing parameter b at ?, X and L points are presented in table 6.

Table5. Phonon frequencies (in cm-1) of Yttrium alloys with As concentration.

Alloys TO(?) LO(?) TA(X) LA(X) TO(X) LO(X) TA(L) LA(L) TO(L) LO(L)
(x=0) YP This work 244.615 260,626 113.063 145.322 251.557 278.860 130.650 213.862 240.155 302.559
Others theoretical data 244.615a
235.69b 260,626a
250.69b 113.063a
116.54b 145.322a
149.72b 251.557a
246.41b 278.860 a
277.90 a 130.650a 213.862a 240.155a 302.559a
(x=0.25) YP0.75As0.25 This work 221.636 278.331 108.311 140.550 220.991 241.191 127.994 204.029 211.538 259.800
Others theoretical data – – – – – – – – – –
(x=0.50) YP0.50As0.50 This work 199.118 251.166 103.521 135.142 209.297 224.741 131.102 185.924 212.321 233.049
Others theoretical data – – – – – – – – – –
(x=0.75) YP0.25As0.75 This work 186.606 233.609 98.435 128.406 202.506 215.213 133.943 172.597 212.871 213.324
Others theoretical data – – – – – – – – – –
(x=1) YAs This work 177. 79 218.362 89.768 116.024 177.685 188.908 122.254 148.467 183.880 199.363
Others theoretical data 177.79 a
175.33b 218.362a 89.768a 116.024a 177.685a 188.908a 122.254a 148.467a 183.880a 199.363a
aRef 15, bRef 16.

Table6: The predicted bowing parameter (in units of cm-1) at the high symmetry points ?, X and L.

Alloy ? point X point L point
YP(1-x)Asx 48.09 46.68 8.438 17.876 21.296 36.572 18.6 19.03 -1.212 71.648
From the table6, we can see that it has the highest bowing values for almost all modes. It is worth noting that the LO–TO splitting, which characterizes the long range electrostatic forces on the vibrating lattice, decreases when the As concentration increases. This indicates that the As bond is less ionic and a better covalence.

3.3.2. Dielectric and born effective charge
Using DFPT method with GGA approximation, we computed the Born effective charges for anions and cations and the electronic and static dielectric constants at ? point and for different x. the results of these properties are listed in the table 7 and plotted in the figures 6.

Table7. Born effective charge, static and electronic dielectric constants.

Alloys with As concentration (x) parameters This work (VCA) Other. Expt
(x=0) YP Z* 3.2111 – –
?(0) 21.4498 – –
?(?) 13.2148 – –
(x=0.25) YP0.75As0.25 Z* 3.1873 – –
?(0) 21.2821 – –
?(?) 13.4949 – –
(x=0.50) YP0.50As0.50 Z* 3.18280 – –
?(0) 21.5711 – –
?(?) 13.5571 – –
(x=0.75) YP0.25As0.75 Z* 3.1703 – –
?(0) 21.8277 – –
?(?) 13.9282 – –
(x=1) YAs Z* 3.1523 – –
?(0) 22.1456 – –
?(?) 14.3737 – –

Figure.6. Born effective charge and dielectric constant as function of As concentration.
The obtained values of born effective charge and dielectric constants are fitted by quadratic relations as follow:
ZAB(1-x)Cx*=xZAC*+1-xZAB*-bZx(1-x) ……………………….. (4)
?AB(1-x)Cx=x?AC+1-x?AB-b?x(1-x)………………………… (5)
Where bZ=-0.001 The bowing parameter of ?(0) and ?(?) are 0.906 and 0.948 respectively. It can be seen that the computed electronic dielectric constant increases within increasing of the As concentration (x). This trend of variation of ? with concentration is consistent with that of the gap the closure of the gaps induces an increase of the electronic dielectric constant and it means that the atoms polarized more with the increase of As composition.

Using pseudopotentials calculations with ABINIT code and using the GGA approximation we have determined the structural and phase transition, elastic and lattice dynamical properties of YP(1-x)Asx ternary alloys in the NaCl structure. For the binary compounds YP and YAs, at x=0 and 1, our GGA calculations are in a reasonable agreement with the available theoretical and experimental data. The calculated elastic constants, phonon frequencies at ?, X and L points, the Born effective charge, the electronic and static dielectric constants have a quadratic form with Arsenic composition. The bowing parameters are predicted. The super cell calculations for the structural and elastic constants give results very close to the VCA ones. Thus, it may be reasonably safe to think that the present calculations predict the elastic and vibrational properties for the YP(1-x)Asx ternary alloys, where the experimental data are not available.

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