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Elastic and thermophysical properties of BAs under high pressure and temperature
Accepted Manuscript
Elastic and thermophysical properties of BAs under high pressure and temperature Salah Daoud , Nadhira Bioud , Noudjoud Lebga PII: DOI: Reference:
S0577-9073(18)31425-4 https://doi.org/10.1016/j.cjph.2018.11.018 CJPH 712
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
14 October 2018 11 November 2018 23 November 2018
Please cite this article as: Salah Daoud , Nadhira Bioud , Noudjoud Lebga , Elastic and thermophysical properties of BAs under high pressure and temperature, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.11.018
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Highlights Ground state of BAs material was obtained; The phase transition pressure and the volume contraction were determined; Elastic constants and sound velocity of BAs material were examined; Thermophysical properties of BAs under high pressure and temperature were obtained.
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Elastic and thermophysical properties of BAs under high pressure and temperature Salah Daoud 1*, Nadhira Bioud 2, 3, Noudjoud Lebga 2, 3 Laboratoire Matériaux et Systèmes Electroniques (LMSE), Université Mohamed Elbachir El Ibrahimi de
Bordj Bou Arreridj, Bordj Bou Arreridj, 34000, Algérie
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Laboratoire d'Optoélectronique et Composants, Université Ferhat Abbas Sétif 1, Sétif, 19000, Algérie.
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Faculté des sciences et de la technologie, Université Mohamed Elbachir El Ibrahimi de Bordj Bou
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Arreridj, Bordj Bou Arreridj, 34000, Algérie.
Abstract
The pseudopotential plane-wave approach in the framework of the density functional theory, and
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the density functional perturbation theory with the generalized gradient approach for the exchange-correlation functional has been used to calculate the structural phase stability, elastic
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constants and thermodynamic properties of boron-arsenide (BAs) compound. The BAs compound
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transforms from the zincblende phase to rock-salt structure; the phase transition pressure was found to be 141.2 GPa with a volume contraction of around 8.2 %. The thermodynamic properties
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under high pressure and temperature up to 125 GPa and 1200 K respectively were also determined, analyzed and discussed in comparison with other data of the literature. The systematic
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errors in the static energy were corrected using the bpscal EEC method. Our results agree well with those reported in the literature, where for example, our calculated melting temperature (2116 K) deviates from the theoretical one (2132.83 K) with only 0.8 %, and the deviation between our result (1.86) of the Grüneisen parameter and the theoretical one (1.921) is only around 3.2 %.
* Corresponding author. E-mail: [email protected]
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Keywords: Phase transition; Thermodynamic properties; Boron arsenide; High-temperature; High-pressure. PACS: 62.50.-p; 64.70.K-; 61.66.Fn; 65.40.-b,
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1. Introduction Boron based pnictides compounds have started gaining attention in recent time due to their tremendous technological promise due to low ionicity, short bond length, high hardness, high melting point and wide band gap [1-6]. The BN compound appears to be more studied
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experimentally and theoretically compared to other ones. Experimentally, the structural phase transition, and the Birch coefficients of the equation of state of BAs compound were determined by Greene et al. [7]. Under high pressure, the low pressure phase is destabilized and structural phase transition occurs. For BAs compound, the phase transition from the zincblende (B3) phase
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to an amorphous state was detected; the limit pressure obtained of the stability of B3 phase is
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around 125 GPa [7]. Whereas the Birch coefficients of the equation of state are: 4.777 Å, 148 ± 6 GPa, and 3.9 ± 0.3 for the equilibrium lattice parameter a0, bulk modulus B0, and the pressure
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derivative of the bulk modulus B0′ respectively [7]. There are other works which reports the first principles calculations and some other methods on
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the study of BAs compound [8-14]. Labidi et al. [15] have performed a first principles calculation
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and reported pressure dependence of electronic and optical properties for BAs compound in B3 structure. They found that the effect of pressure on the main band gaps, show that the band gap min decreases
with increasing of pressure.
Furthermore, the scarce data on the thermodynamic properties under high pressure and temperature motivates us to perform a systematic study of these properties for BAs, which may shed some light on this material for its possible technological applications at high pressure and
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temperature. In the present work, we report on first-principles calculations of the structural parameters, the pressure of the phase transition and the temperature and pressure effect on the thermodynamic properties of BAs in B3 structure using the pseudopotential plane wave method in the framework of the density-functional theory (DFT) [16]. Besides, the elastic properties and
(DFPT) [16] were also obtained, examined and presented.
2. Computational methods
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some other derivative quantities in the framework of the density functional perturbation theory
In the present study, the calculations were carried out using ABINIT code [17], which is based on
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the pseudopotential plane wave (PP-PW) approach in the framework of DFT and DFPT. The interactions between the valence electrons and the nuclei and core electrons were described by Troullier-Martins type norm-conserving pseudopotentials, which have been generated thanks to
gradient approach (GGA) [19].
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the FHI98PP code [18]. The exchange-correlation energy was evaluated in the generalized
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A set of convergence tests was performed in order to correctly choose the mesh of k-points and the cutoff kinetic energy in the plane waves to start the ground state calculations. In order to find an
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appropriate energy cutoff, the total energy was calculated as a function of energy cutoff. The electronic wave functions are represented in a plane wave basis set with a kinetic energy cutoff of
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60 Ha. The Brillouin zone was sampled by (6x6x6) Monkhorst and Pack mesh of k points [20].
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We have also checked more dense grids but the results did not changed considerably. Convergence tests prove that the Brillouin zone sampling (6x6x6) and the kinetic energy cutoff 60 Ha were sufficient to guarantee a good convergence. In order to obtain the pressure and temperature dependence of the thermodynamic properties of BAs compound, the Debye model [21, 22] was successfully applied; our calculations are implemented through the Gibbs code [22]. The Gibbs code is used to obtain much thermodynamic quantities from the minimization of Energy-Volume (E, V) data, which is in our case obtained
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from the ABINIT code [17]. Gibbs implements several models for the inclusion of temperature effects to the results of an ab-initio calculation. The Debye model requires only the input of the static Energy-Volume (E, V) data and optionally the experimental Poisson’s ratio [21, 22]. So through the Debye model, one could calculate several thermodynamic quantities including the volume V, isothermal bulk modulus BT, etc. For more detail on the calculation of the
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thermodynamic properties of materials and the different relationships used here, please see the Refs. [21-24].
3. Results and discussion
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3.1. Equilibrium structural parameters
Usually, the structural equilibrium parameters can be predicted from Ab-initio calculation, using the pressure - unit cell volume (P-V) data, or the energy - unit cell volume (E-V) data. In order to obtain the structural parameters of BAs compound in both cubic zincblende (B3) and sodium
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chloride (B1) structures the energy - unit cell volume (E-V) data was used. The equilibrium lattice
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constant a0, bulk modulus B0 and pressure derivative of bulk modulus B0′ have been computed by minimizing the total energy by means of Murnaghan’s equation of state (EoS) [25] V / V B0 BV 1 '0 0 0' B0 1 B0 1 '
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BV E V E V0 0 ' B0
(1)
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Where V0 is the equilibrium volume, V is the volume, and E (V0) is the equilibrium energy.
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Fig. 1 presents the variation of the total energy as a function of the unit cell volume of BAs compound in both B3 and B1 structures. Our calculated values of a0, B0, and B0′ are summarized in Table 1 and compared with other theoretical results [1-6] and experimental data [7]. From the results summarized in Table 1, it is observed that our calculated values of a0 in both structures are in good agreement compared to available data [1-7]. It is important to note that the lattice parameter is one of the fundamental structural quantities for a solid directly related to bond length,
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which is determined by the density of valence electrons between the atoms of crystal. BAs compound is one of boron pnictides materials which in general have smaller unit cell volume and larger bulk modulus comparatively to some other conventional III-V compounds. For B3 structure, our obtained value (4.8342 Å) of a0 overestimate the experimental one (4.777 Å) [7] by only less than 1.2 %.
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Our calculated values of B0 and B0′ of both structures are generally also in agreement with the available theoretical data [1-6]. Where for example, our obtained value (126.74 GPa) of B0 of B3 structure underestimate the theoretical value (131 GPa) calculated by El Haj Hassan [3] by only
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about 3.25 %. 3.2. Pressure of the phase transition
The first method used to predict the threshold pressure of the phase transition is the dynamic calculation (phonon) at different pressures, which can find the dynamic instability (negative
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frequencies) and therefore the corresponding transition pressure. The second is from the
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thermodynamic calculation, the phase transition occurs when a change in the structure appears, which is caused by the variation of the free energy. The change in free energy becomes zero at
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phase transition pressure. However, the stability of any particular structure corresponds to the lowest Gibbs free energy, which is given by [8, 24]. (2)
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G = U + PV - TS
Where: U is the internal energy, P the pressure, T the temperature, S the entropy and V is the
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volume.
To investigate the pressure-induced structural transition of BAs, we have optimized at T = 0 K the cell parameters and atomic positions for both B3 and B1 phases. At T = 0 K, the Gibbs free energy becomes equal to the enthalpy [8, 24] H = U + PV
(3)
The computed enthalpies versus pressure curves for both phases of interest are shown in Fig. 2.
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From Fig. 2, we can see that the Gibbs’ free energy for B3 phase is more negative than this of B1 phase. The enthalpy versus pressure curves corresponding to B3 and B1 phases cross at a pressure of 141.2 GPa, suggesting therefore that the transition pressure (Pt) from B3 to B1 is 141.2 GPa. For comparison, our value obtained in the present work is listed in Table 2, along with the available experimental [7], and other theoretical ones [6, 8, 11-14, 26, 27].
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During the analysis of the phase transition from B3 to B1 phase, we also examined the reduction in the volume during this transition. The equation of state curves of BAs compound (plotted between V(P)/V(0) and pressure) for both B3 and B1 phases obtained in this work are plotted in Fig. 3. From these curves, one can estimate the volume collapse at the point of transition. The
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value of the volume collapse for BAs compound computed in this work is about 8.2 %. For comparison, our value obtained in the present work is listed also in Table 2 along with previous theoretical data [8, 11, 14, 26-28]. It is notable that our calculated value of the volume collapse is
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in good agreement compared to theoretical ones (9 %) obtained by Wentzcovitch and Cohen [14], (9.93 %) obtained in our previous work [8], and (7.92 %) obtained by Sarwan et al. [26]. Our
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obtained value (8.2 GPa) of the volume collapse overestimate the theoretical one (7.92 %) [26] by only about 3.54 %; so our result of the volume collapse is localized between these two theoretical
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ones. We note also that Cui et al. [11] found the phase transition B3 → B1 occurs at around 134
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0 %).
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GPa, but with no changing in the volume of the unit cell during this transition (Volume collapse =
3.3. Elastic properties The elastic properties of material define the mechanical behavior of solid that undergoes stress, deforms and then returns to its original shape after stress ceases [27]. The DFPT method was recently widely applied to calculate several physical properties, such as dynamic, piezoelectric and elastic properties of materials. The calculations of the elastic constants presented in this work were carried out using the ABINIT code [17] in the framework of the DFPT (For more detail on the
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calculation of the elastic constants with the DFPT method, please see for example our previous work [8]). For crystals with cubic structures, the matrix of the elastic constants contains only 3 independent elastic constants (C11, C12 and C44), the aggregate bulk modulus in this case is given as [8]: B (C11 2C12 ) / 3
(4)
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The evaluated elastic constants Cij, and the bulk modulus B of BAs at zero-pressure are presented in Table 3, along those of the literature [2, 6, 8, 10, 13, 26]. From the Table 3, we observe that our calculated elastic constants Cij are in general in reasonably good agreement with other theoretical ones [2, 6, 8, 10, 13, 26], where for example, the value (264.53 GPa) of C11 obtained by us is
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localized between the two theoretical values (275 GPa) and (252 GPa) obtained by Meradji et al. [2] and Thakore et al. [13] respectively. The value (62.88 GPa) of C12 obtained by us deviates from the theoretical one (63 GPa) reported by Meradji et al. [2] with only about 0.2 %.
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On the macroscopic scale, the elastic properties were usually described using the elastic moduli (Young modulus E, Poisson's ratio σ,…etc), which are varied to the direction of the single crystal.
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The Young's modulus E is a measure of the ability of solids to defend longitudinal stress [29]. For
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the aggregate polycrystalline material, the Young modulus E and Poisson's ratio σ are calculated with the following equations [29]:
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E 9B G/ (3B G) , and σ (3B 2G) / 2(3B G)
(5)
Where: B is the bulk modulus and G is the isotropic shear modulus.
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Our calculated values of G, E and σ are presented in Table 3, and compared with other available theoretical data of the literature [6, 10, 13, 26]. It is seen also that, our calculated values of G, E and σ are in general in good agreement with the previously calculated data [6, 10, 13, 26], where for example, the value (0.14) of σ obtained by us is localized between the two theoretical values (0.146) and (0.136) obtained in our previous work [8] and by Sarwan et al. [26] respectively. In addition our value of σ is equal exactly that reported by Bing et al. [6]. As we see in Table 3, BAs
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compound has a large value of G which implies a high level of resistant of this compound against the shear strain. The anisotropy factor A is quantity, which indicts the degree of the elastic anisotropy in material. For a completely isotropic material, the value of A is a unity (A = 1) [26, 29]. If the value of A differs than 1, so the solid shows the degree of elastic anisotropy. It can be calculated as follow
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[30, 31] : A 2C44 / (C11 C12 ) . The calculated value of A for BAs is around 1.43, it is greater than 1 which indicates that our material of interest shows elastic anisotropic character. Our obtained value of the anisotropy factor A is in good agreement compared to the values 1.445 and 1.461 reported by Wang and Ye [10] and Sarwan et al. [26]. The deviations between our value of A and
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the values reported by Wang and Ye [10] and Sarwan et al. [26] are around 1.04 % and 2.12 % respectively. If A <1, the crystal is stiffest along <100> cube axes, and when A >1, it is stiffest along the <111> body diagonals [30, 31].
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For cubic diamond and cubic zinc-blende crystals, the Kleinman internal strain parameter is quantity which describing the relative position of the cation and anion sub-lattices. In the central
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force model, this quantity can be expressed as function of the elastic constants C11 and C12 as
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follow [5] (C11 8C12 ) / (7C11 2C12 )
(6)
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It is known that a low value of ζ implies there is a large resistance against bond bending or bondangle distortion and vice versa [10]. Our obtained value of ζ of BAs is equal to 0.388; it is in
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excellent agreement with the theoretical value (0.39) reported by Ustundag et al. [5]. The deviation between our value and the value reported by Ustundag et al. [5] is only around 0.5 %. Our value of ζ is slightly higher than the theoretical value 0.362 reported by Wang and Ye [10]. For cubic polycrystalline materials, the average sound velocity vm is usually used to study the acoustic wave propagation; this quantity is given by the following expression [8, 26, 29]
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vm 1/3 (2/vt 3 ) (1/vl 3 )
10
1/3
(7)
Where: vl and vt are the longitudinal and transverse elastic wave velocities respectively, which are determined by using the following expressions [8, 29]: vl
3B 4G / 3 , and
vt G /
(8)
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where ρ is the crystal density. The calculated longitudinal (vl), transverse (vt) and average (vm) sound velocities of BAs
compound are: 7669 m/s, 4975 m/s and 5458 m/s respectively. They are presented in Table 4, and compared with other available theoretical data of the literature [6, 8, 13, 26, 27, 32]. Our results of
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vl, vt and vm are in general in very good agreement compared with our previous work [8] and with the results reported by Ustundag et al. [5], where for example the deviation between our value (5458 m/s) of vm and the value (5490 m/s) reported in our previous work [8] is only around 0.6 %.
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From Table 4, we can see that the values of vl, vt and vm obtained previously by Sarwan et al. [26] are very higher than our values and other data of the literature.
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One of the standard methods largely used to estimate the Debye temperature θD of solid is from the elastic constants and sound velocity data; it can be calculated from the average sound velocity
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vm using the following formula [8]:
(9)
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θD h/k B 3/4 Va 1/3vm
Where: h is the Planck constant; kB is the Boltzmann constant, and Va the atomic volume.
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The calculated Debye temperature θD of BAs at zero-pressure is about 672 K. It is listed in Table 4 and compared with available theoretical [5, 8, 26, 27] and experimental [32] data. It can observed that our calculated Debye temperature θD is slightly higher than the experimental one (625 K) [32], but in general it is in good agreement with other theoretical ones of the literature [5, 8], where for example, the deviation between our value (672 K) and the theoretical one (693.09 K) reported by Ustundag et al. [5] is only around 3 %.
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The melting temperature Tm (expressed in K) of several solids with cubic structure can be estimated from the elastic constant C11 (expressed in GPa) according to the following empirical relationship [8] Tm 553 5.91 C11
(10)
The calculated melting temperature Tm of BAs at zero-pressure is about 2116 K. It is listed in
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Table 4 and compared with available theoretical [5, 8] and experimental [32] data. It is seen that, our calculated melting temperature Tm is also in good agreement with other data of the literature [5, 8, 32], where for example, the value (2116 K) obtained by us deviates from the theoretical one
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(2132.83 K ) reported by Ustundag et al. [5] with only about 0.8 %. 3.4. Thermodynamic properties
The thermodynamic properties of materials may play an important role in physics of solid state matter [23], the good knowledge of these properties can allow extending our knowledge on their
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specific behavior when undergoing severe constraints of high pressure and temperature [33]. It is
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found that the empirical energy corrections (EECs) are able to reproduce the thermodynamic properties and EoS of solid materials at very high temperature [34]. So, in order to improve the
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systematic deviation of PBE functional used here, and to reproduce the experimental results in the range of the validity of the quasi-harmonic Debye model approximation, the empirical energy
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correction (EEC) becomes necessary. One of the methods used to correct the systematic errors in
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the static energy Esta (V) is that called bpscal EEC [34]. The BPSCAL correction modifies the static energy according to [34] ~
Esta (V ) Esta
B exp Vexp (V0 ) B0 V0
Esta
VV 0 V exp
Esta
(V0 )
(11)
Where (V0, B0) are calculated static equilibrium properties, and (Vexp, Bexp) are experimental ones. The experimental values of the volume Vexp and the bulk modulus Bexp used here to correct the total energy are measured by Greene et al. [7]; they are equal to 183 (a.u)3 and 148 GPa
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respectively. In the present work, the thermal properties are determined in the temperature range from 0 K to 1200 K, and the pressure effect is studied in the range of 0-125 GPa. Figures 4 (a) and 4 (b) display the pressure dependence of V at various temperatures and the temperature dependence of the V at various pressures for BAs compound. Note that for our compound of interest, V decreases
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with increasing pressure at all temperatures of interest. At any given pressure, V increases with raising temperature. The change in the behavior of the volume with pressure is obvious as applied pressure brings the atoms closer and hence it reduces the lattice parameter which in turn leads to the decrease in the volume [23]. On the other hand, when the temperature increases, this leads the
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dilatation of the lattice parameter leading thus to the increase in the volume of our crystal of interest. The similar trend was also observed in the case of BSb semiconducting compound [24]. The bulk modulus is one of the most important mechanical parameters of material; it usually
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decreases as temperature rises. The isothermal bulk modulus BT is given by the following expression [21-23],
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BT = -V(dp/dV)T = V(d 2 F/dV 2 )T
(12)
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where F is the vibrational Helmholtz free energy. Figure 5 displays the pressure and temperature dependence of BT for BAs compound. We observe
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that BT increases with increasing pressure at all considered temperatures. The behavior is
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monotonic for BAs. It should also be noted that for each given pressure BT decreases with raising temperature up to 1200 K. One may thus conclude that the material under study becomes harder with increasing pressure or decreasing temperature. At zero pressure and zero temperature, the obtained value of BT of BAs compound is 149.5 GPa, which is in excellent agreement with the experimental one (148 ± 6 GPa) reported by Greene et al. [7]. Our obtained value of BT overestimates the experimental data (148 GPa) by only around 1 %. At zero pressure, BT increases almost quadratically with increasing pressure; where at 1200 K, it
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becomes equal to 118.5 GPa. The behavior of BT under pressure and temperature is very similar to that found in BSb material [24]. The knowledge of the heat capacity of a material not only provides essential information on its vibrational properties but also is mandatory for many technological applications [35]. The heat capacity at constant volume Cv is obtained using the expression [23, 24]
where U is the total internal energy, and T is the temperature.
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(13)
CV = (dU/dT)V
Figures 6 (a) and 6 (b) show the pressure dependence of Cv at various temperatures and the temperature dependence of Cv at various pressures for BAs compound. It can be seen from Fig. 6
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(b) that Cv increases exponentially with the temperature for T < 500 K. For temperatures higher than 500 K, it follows naturally the Debye model and approaches the DuLong-Petit limit indicating that at high temperature all phonon modes are excited by the thermal energy [23]. It can
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be noted that similar qualitative behavior has been reported for the heat capacity Cv versus temperature for BSb semiconducting compound [24], magnesium silicate (MgSiO3) perovskite
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material [35], superconducting MgCNi3 material [36], and for both wurtzite and zinc-blende GaN
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compound [37]. At zero-pressure and 300 K, the calculated Cv of BAs is around 42 J.mol-1.K-1; it is very higher than the heat capacities of wurtzite and zinc-blende GaN compound, which are 34.4
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and 34.3 J.mol-1.K-1 respectively [37]. The heat capacity at constant pressure Cp can be obtained using the expression: CP (H / T) P
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[24], where: H is the enthalpy. Figure 7 shows the calculated Cp for different pressures and temperatures for BAs compound. It can be seen from Fig. 7 that the heat capacity Cp increases exponentially with the temperature for T < 500 K, and becomes almost stable (especially at very high pressure) for T ˃ 500 K. The similar qualitative behavior has been reported for Cp versus temperature for magnesium silicate (MgSiO3) perovskite [35].
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The effect of temperature and pressure on the entropy S is presented in Fig. 8. At low temperature and zero-pressure, one can note that the entropy increases quickly with raising the temperature T, then the behavior of the entropy as a function of temperature becomes more like a sub-linear behavior. By increasing pressure from 0 up to 125 GPa, the behavior of the entropy versus temperature remains qualitatively almost the same. However, from the quantitative point of view,
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the entropy seems to decrease slightly in magnitude with increasing pressure for BAs. This is in agreement with the variation of S with pressure and temperature deduced for BSb compound [24]. Figures 9 (a) and (b) show the pressure dependence of the Debye temperature θD at various temperatures and the temperature dependence of θD at various pressures for BAs compound. It can
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be seen from Fig. 9 that the Debye temperature increases monotonically with increasing pressure. At a fixed pressure, the Debye temperature decreases with raising temperature for BAs compound. At zero pressure and zero temperature, the obtained value of θD for BAs compound is around
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570.5 K. This value is slightly lower than the experimental one (625 K) reported in Ref. [32]. Almost our obtained value of θD is slightly lower than the measured value (625 K) [32], it can
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observed that, our calculated value of θD is better than the theoretical ones (511.53 K) and (329.19 K) reported by Sarwan et al. [26] and Varshney et al. [27] respectively.
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In the quasi-harmonic Debye model approximation [21, 22], the Grüneisen parameter can be
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given by the following expression [21-23],
d ln D (V ) / d ln V
(14)
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Where V is the volume and D (V ) is the Debye temperature. Figure 10 displays the pressure and temperature dependence of the Grüneisen parameter of BAs compound. As can be seen from Fig. 10, the Grüneisen parameter for BAs decreases with increasing pressure and increases with temperature. At zero pressure and ambient temperature, our obtained value of the Grüneisen parameter is around 1.86. This value is in good agreement
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compared to the theoretical one (1.921) reported by Varshney et al. [27]. The deviation between our value (1.86) and the theoretical one (1.921) reported in Ref. [27] is only around 3.2 %. The thermal expansion coefficient α is a quantity which is used for describing the temperature effect on the size of object (dimensions change with a change in temperature) [38]. The volumetric thermal expansion coefficient α and the Grüneisen parameter are related as follow [23]: γ CV /BTV
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(15)
In Figure 11, we plot α as a function of temperature and pressure for BAs compound. The
behavior of this quantity is similar to that observed for the constant volume heat capacity and the
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entropy. At zero-pressure and T = 300 K, our obtained value of α is around 3.27 x 10-5 K-1. Our calculations show that α increases monotonically with increasing temperature and decreases with increasing pressure. This behavior can be explained that the anharmonic effect becomes less important at high pressures. With the increase of pressure, the volume of the solid reduces and the
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atoms come closer to each other, increasing the depth of the potential energy well and reducing the anharmonic nature of the potential energy curve at high temperatures [35]. It can note that a
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similar qualitative behavior has been reported for α versus temperature for BSb [24]
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semiconducting compound, and for magnesium silicate (MgSiO3) perovskite material [35]. Since thermal expansivity of material is a result of anharmonicity in the potential energy; this quantity
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[35].
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becomes less dependent of temperature and pressure in the high temperature and pressure domain
4. Conclusion In summary, we have investigated the EoS parameters, the phase transition pressure, the elastic properties, and many other thermodynamical variables of BAs in B3 structure using density functional theory. We have calculated the ground state properties like lattice parameter, bulk modulus and first order pressure derivative of bulk modulus which are found in good agreement
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with the earlier available results. Our static calculations for BAs reveal that this compound follows B3-B1 phase transition. The phase transition pressure from B3 to B1 phase of the BAs compound has been determined using the total energy, and the Gibbs free energy. Our obtained value of 141.2 GPa is in general in agreement with other theoretical data. The unit cell volume, the isothermal bulk modulus, the heat capacity, the Debye temperature, the
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Grüneisen parameter and some other thermodynamic properties under high pressure up to 125 GPa and temperature up to 1200 K were also determined, analyzed and discussed. Our results agree well with those reported in the literature, where for example, the deviation between our result (1.86) of the Grüneisen and the theoretical one (1.921) is only around 3.2 %. At high
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temperatures, CV converges to a near-constant value (DuLong-Petit limit), while S, CP and θD increase monotonously with the temperature. All thermodynamic properties our compound of interest were found vary monotonically with either temperature or pressure.
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BSb, J. Phys. Condens. Mat. 12 (2000) 5655- 5668. [2] H. Meradji, S. Drablia, S. Ghemid, H. Belkhir, B. Bouhafs, A. Tadjer, First-principles elastic constants and
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electronic structure of BP, BAs, and BSb, Phys. Stat. Sol. (b) 241(13) (2004) 2881-2885. [3] F. El Haj Hassan, First‐principles study of BNxSb1–x, BPxSb1–x and BAsxSb1–x alloys, Phys. Stat. Sol. (b) 242 (2005) 3129-31.37.
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[4] A. Zaoui, F. El Haj Hassan, Full potential linearized augmented plane wave calculations of structural and electronic properties of BN, BP, BAs and BSb, J. Phys. Condens. Mat. 13 (2001) 253- 262. [5] M. Ustundag, M. Aslan, Battal G. Yalcin, The first-principles study on physical properties and phase stability of
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Boron-V (BN, BP, BAs, BSb and BBi) compounds, Comput. Mater. Sci. 81 (2014) 471-477. [6] L. Bing, L. R. Feng, Y. Yong, Y. X. Dong, Characterisation of the high-pressure structural transition and elastic properties in boron arsenic, Chin. Phys. B. 19 (7) (2010) 076201. [7] R. G. Greene, H. Luo, A. L. Ruoff, S. S. Trail, F. Jr. Disalvo, Pressure induced metastable amorphixation of BAs: evidence for a kinetically frustrated phase transformation, Phys. Rev. Lett. 73 (18) (1994) 2476-2479. [8] S. Daoud, N. Bioud, N. Bouarissa, Structural phase transition, elastic and thermal properties of boron arsenide: Pressure-induced effects, Mater. Sci. Semicond. Process. 31 (2015) 124-130. [9] M. J. Herrera‐Cabrera, P. Rodríguez‐Hernández, A. Muñoz, First‐principles elastic properties of BAs, Int. J. Quantum Chem. 91 (2003) 191-196.
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[10] S. Q. Wang, H. Q. Ye, First-principles study on elastic properties and phase stability of III-V compounds, Phys. Stat. Sol. (b) 240 (2003) 45-54. [11] S. Cui, W. Feng, Z. Feng, Y. Wang, First-principles study of zinc-blende to rocksalt phase transition in BP and BAs, Comput. Mater. Sci. 44(4) (2009) 1386-1389. [12] F. El Haj Hassan, H. Akbarzadeh, M. Zoaeter, Structural properties of boron compounds at high pressure J. Phys: Condens. Mat. 16(3) (2004) 293. [13] B.Y. Thakore, M. J. Joshi, N. K. Bhatt, A.R. Jani, High pressure phases of Boron compounds using first principles approach, J. Optoelectron. Adv. M. 11 (2009) 461- 465.
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[14] R. M. Wentzcovitch, M.L. Cohen, Theoretical study of BN, BP, and BAs at high pressures, Phys. Rev. B 36 (1987) 6058.
[15] S. Labidi, H. Meradji, S. Ghemid, S. Meçabih, B. Abbar, Pressure dependence of electronic and optical properties of zincblende BP, BAs and BSb compounds, J. Optoelectron. Adv. M. 11 (2009) 994-1001.
[16] E. Engel, R.M. Dreizler, Density Functional Theory, Springer-Verlag Berlin Heidelberg, New York, (2011). [17] The ABINIT code is a common project of the Université Catholique de Louvain, Corning Incorporated, and other
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contributors. http://www.abinit.org
[18] M. Fuchs, M. Scheffler, Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density- functional theory, Comput. Phys. Commun. 119(1) (1999) 67-98.
[19] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett 77(18) (1996) 3865- 3868.
[20] H. J. Monkhorst, J. D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B 13 (1976) 5188.
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[21] M.A. Blanco, E. Francisco, V. Luaña, GIBBS: isothermal-isobaric thermodynamics of solids from energy curves using a quasi-harmonic Debye model, Comput. Phys. Commun. 158 (2004) 57.
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[22] A. Otero-de-la-Roza, V. Luaña, GIBBS 2: A new version of the quasiharmonic model code. II. Models for solid-state thermodynamics, features and implementation, Comput. Phys. Commun. 182 (2011) 2232. [23] N. Bioud, K. Kassali, N. Bouarissa, Thermodynamic properties of compressed CuX (X = Cl, Br) compounds: Ab
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initio study, J. Electron. Mater. 46 (2017) 2521-2528. [24] N. Bioud, X. W. Sun, S. Daoud, T. Song, Z. J. Liu, Structural stability and thermodynamic properties of BSb
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[26] M. Sarwan, P. Bhardwaj, S. Singh, Zinc-blende to rock-salt structural phase transition of BP and BAs under high pressure, Chem. Phys. 426 (2013) 1-8. [27] D. Varshney, G. Joshi, M. Varshney, S. Shriya, Pressure induced mechanical properties of boron based pnictides, Solid. State. Sci. 12(5) (2010) 864-872. [28] G. J. Ackland, High-pressure phases of group IV and III-V semiconductors, in High-Pressure Surface Science and Engineering (Y. Gogotsi and V. Domnich, Eds), Chap 2, 3, IOP Publishing Ltd, Bristol and Philadelphia (2004) p. 142. [29] Md. Zahidur Rahaman, Md. Lokman Ali, Md. Atikur Rahman, Pressure-dependent mechanical and thermodynamic properties of newly discovered cubic Na 2He, Chin. J. Phys. 56 (2018) 231-237.
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[30] R. E. Newnham, Properties of Materials: Anisotropy, Symmetry, Structure (Oxford: Oxford University Press) (2005) [31] N. Bioud, X.W. Sun, N. Bouarissa, S. Daoud, Elastic constants and related properties of compressed rocksalt CuX (X = Cl, Br): Ab initio study, Z. Naturforsch. A 73 (8) (2018) 767-773. [32] D.R. Lide, Handbook of Chemistry and Physics, 80 th ed., CRC Publication, OCLC World Cat, (1999-2000). [33] P.K. Jha, Phonon spectra and vibrational mode instability of MgCNi 3, Phys. Rev. B 72 (2005) 214502. [34] F. Luo, Y. Cheng, L-C. Cai, X-R. Chen, Structure and thermodynamic properties of BeO: Empirical corrections in the quasiharmonic approximation, J. Appl. Phys. 113 (2013) 033517.
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[35] Z-J. Liu, X-W Sun, C-R. Zhang, J-B Hu, T. Song, J-H. Qi, Elastic tensor and thermodynamic property of magnesium silicate perovskite from first-principles calculations, Chin. J. Chem. Phys. 24 (6) (2011) 703-710. [36] P. K. Jha, Sanjay D. Gupta, S. K. Gupta, Puzzling phonon dispersion curves and vibrational mode instability in superconducting MgCNi3, AIP Advances 2 (2012) 022120.
[37] H. Qin, X. Luan, C. Feng, D. Yang , G. Zhang, Mechanical, thermodynamic and electronic properties of wurtzite and zinc-blende GaN crystals, Materials 10 (2017) 1419 (15 pages).
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[38] M. Xing, B. Li, Z. Yu, and Q. Chen, Elastic anisotropic and thermodynamic properties of I-4m2-BCN, Acta
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Phys. Pol. A 129 (2016) 1124-1130.
Figure Captions:
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Figure 1: Total energy versus volume of BAs compound in both B3 and B1 structures.
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-258
-261 -262
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-260
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Energy (eV)
-259
Zinc-blende phase Rock-salt phase
-263 -264
16
20
24
28
32 3
Volume (Å )
36
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B1 B3
-250
-255
Pt = 141.2 GPa
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-265 0
30
60
90
120
150
180
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Pressure (GPa)
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Entalpy (eV)
-245
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Figure 3: Variations of the relative volume (Vp/V0) of B3 and B1-BAs versus pressure. 1.00
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0.95 0.90
0.75 0.70
(B3)
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0.80
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Vp/V0
0.85
0.65 0.60
(Vp(B3)-Vp(B1))/V0 = 8.2 %
(B1)
0.55 0.50
0
40
80
120
Pressure (GPa)
160
200
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ACCEPTED MANUSCRIPT Figure 4: (a) Volume versus pressure at different temperatures of BAs compound, (b) Volume versus temperature at different pressures of BAs compound.
T= 0 K T= 300 K T= 800 K T= 1200 K
(a)
28
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Volume (Å )
26
22 20 18
0
25
50
75
100
125
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P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
27.0
(b)
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Volume (Å )
25.5
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24.0 22.5
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21.0
18.0
0
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19.5
300
600
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Temperature (K)
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900
1200
20
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Figure 5: (a) Isothermal bulk modulus BT versus pressure at different temperatures of BAs compound, (b) BT versus temperature at different pressures of BAs compound.
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T= 0 K T= 300 K T= 800 K T= 1200 K
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(a)
360 300 240 180 120 0
25
50
75
100
125
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540 480
360
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P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
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300
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BT (GPa)
420
180
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120 60
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BT (GPa)
420
0
300
600
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Temperature (K)
900
1200
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42 40
T= 300 K T= 600 K T= 800 K T= 1200 K
(a)
38 36 34 32 30 0
25
50
75
100
125
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Pressure (GPa) 50
30
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P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
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(b)
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Cv (J/mol K)
40
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10
0
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Cv (J/mol K)
44
0
300
600
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Temperature (K)
900
1200
22
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Figure 7: (a) Constant pressure heat capacity CP versus pressure at various temperatures of BAs compound, (b) CP versus temperature at various pressures of BAs. 56 52
44 40
T= 300 K T= 600 K T= 800 K T= 1200 K
(a)
36 32 28
0
25
50
75
100
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Pressure (GPa) 56 48
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32 24
(b)
16
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Cp (J/mol K)
40
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8 0
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Cp (J/mol K)
48
0
300
600
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Temperature (K)
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
900
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(a)
T= 300 K T= 600 K
T= 800 K T= 1200 K
75 60 45 30 15
0
25
50
75
100
125
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105 90
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60 45 30
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S (J/mol K)
75
(b)
PT
15 0
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S (J/mol K)
90
0
300
600
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Temperature (K)
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
900
1200
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935 850
T= 0 K T= 300 K T= 800 K T= 1200 K
(a)
765 680 595 510
0
25
50
75
100
125
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Pressure (GPa)
935 850
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
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765
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680 595 510
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Debye temperature (K)
1020
(b)
0
300
600
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Temperature (K)
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Debye temperature (K)
1020
900
1200
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T= 0 K T= 300 K T= 800 K T= 1200 K
(a)
1.89 1.80 1.71 1.62 1.53 1.44 1.35 1.26 0
25
50
75
100
125
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Pressure (GPa)
1.89
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
1.80
(b)
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1.62 1.53
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Grüneisen parameter
1.98
1.71
1.44 1.35
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1.26 0
300
600
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Temperature (K)
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Grüneisen parameter
1.98
900
1200
26
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Figure 11: (a) Volumetric thermal expansion coefficient versus pressure at different temperatures of BAs compound, (b) Volumetric thermal expansion coefficient versus temperature at different pressures of BAs.
T= 300 K T= 600 K T= 800 K T= 1200 K
(a)
4,50
3,00 2,25 1,50 0,75 0
25
50
75
100
4,8 4,0
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-5
(10 /K)
2,4 1,6
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0,8 0,0
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
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(b)
3,2
125
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0
300
600
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Temperature (K)
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(10 /K)
3,75
900
1200
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Table Captions: TABLE 1: Equilibrium lattice parameter, bulk modulus, and its pressure derivative of BAs compound in both cubic zincblende and cubic sodium chloride structures.
B1
Table 1:
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parameter
work
a0 (Å)
4.8342
4.728 a, 4.812 b, 4.743 c, 4.814d, 4.784 e, 4.812 f , 4.779g, 4.777
B0 (GPa)
126.74
144 a, 133 b, 152c, 131d, 137 e, 130.91f , 138.29 g, 148 ± 6 h
B'0
3.58
4.00 a, 3.75 b, 3.65c, 3.49 e, 3.71 f , 4.09 g, 3.9 ± 0.3 h
a0 (Å)
4.6168
4.611 b, 4.534 c, 4.619 e, 4.622 f , 4.581 g
B0 (GPa)
114.56
132 b, 158c, 135e, 125.18f , 142.88 g
B'0
3.88
3.76 b, 3.55c, 3.44 e, 2.976f , 3.825 g
h
, a Ref. [1], b Ref. [2] for GGA calculation, c Ref. [2] for LDA calculation, d Ref. [3], e Ref. [4], f Ref. [5],
Ref. [7] Exp.
g
Ref. [6],
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h
Present
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B3
EoS
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Phase
TABLE 2: Transition pressure from B3 to B1 structure, and the corresponding volume contraction for
Parameter Pt (GPa)
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PT
BAs semiconducting compound.
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141.2
113.42 a, 125 b, 125 c, 134 d, 98 e, 110 f, 110 g , 93 h, 110 i
8.2
9.93c , 0 d, 9 g , 7.92 h, 4.2 i, 15 j
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ΔV/V0 (%)
Present work
Table 2: a Ref. [6], b Ref. [7] Exp,
c
Ref. [8],
d
Ref. [11], e Ref. [12], f Ref. [13],
g
Ref. [14], h Ref. [26], i Ref. [27] , j Ref. [28].
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TABLE 3: Elastic constants of BAs semiconducting compound in cubic zincblende structure.
Present work
Other Works
C11 (GPa)
264.53
275 a, 295 b, 286.39 c, 301.26 d, 291.4 e, 252 f, 289 g
C12 (GPa)
62.88
63 a, 78 b, 70.96c, 77.23 d, 72.8 e, 78 f, 70 g
C44 (GPa)
143.89
150 a, 177 b, 57.50 c, 163.87 d, 157.9 e, 128 f, 160 g
B (GPa)
130.10
133 a, 152 b, 142.77 c, 151.91 d, 138 e, 148 f, 143 g
G (GPa)
124.78
135.26 c,
E (GPa)
283.65
308.38 c, 262.3 e, 312 g
σ
0.14
0.14 c, 0.146 d, 0.2 e, 0.136 g c
Ref. [6], d Ref. [8], e Ref. [10], f Ref. [13], g Ref. [26].
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Table 3: a Ref. [2] for GGA calculation, b Ref. [2] for LDA calculation,
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Parameter
This work
(m/s)
7669
(m/s)
4975
vt
(m/s) vm
Tm (K)
7779.69 a 7373b, 15816.03 c, 8350 d 5114.69 a 5039 b , 8188 c, 4500 d
5458
5601.24a 5490 b, 10884.2c, 6023 d,
672
693.09 a, 698.16 b, 511.53c, 329.19 d, 625 e
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θD (K)
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vl
Others
PT
Parameter
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TABLE 4: Sound velocity, Debye temperature and melting temperature of BAs compound.
2116
2132.83 a, 2333.5 b, 2300 e
Table 4: a Ref. [5], b Ref. [8], c Ref. [26], d Ref. [27],
e
Ref. [32] Exp.
Elastic and thermophysical properties of BAs under high pressure and temperature Salah Daoud , Nadhira Bioud , Noudjoud Lebga PII: DOI: Reference:
S0577-9073(18)31425-4 https://doi.org/10.1016/j.cjph.2018.11.018 CJPH 712
To appear in:
Chinese Journal of Physics
Received date: Revised date: Accepted date:
14 October 2018 11 November 2018 23 November 2018
Please cite this article as: Salah Daoud , Nadhira Bioud , Noudjoud Lebga , Elastic and thermophysical properties of BAs under high pressure and temperature, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.11.018
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Highlights Ground state of BAs material was obtained; The phase transition pressure and the volume contraction were determined; Elastic constants and sound velocity of BAs material were examined; Thermophysical properties of BAs under high pressure and temperature were obtained.
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Elastic and thermophysical properties of BAs under high pressure and temperature Salah Daoud 1*, Nadhira Bioud 2, 3, Noudjoud Lebga 2, 3 Laboratoire Matériaux et Systèmes Electroniques (LMSE), Université Mohamed Elbachir El Ibrahimi de
Bordj Bou Arreridj, Bordj Bou Arreridj, 34000, Algérie
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1
2
Laboratoire d'Optoélectronique et Composants, Université Ferhat Abbas Sétif 1, Sétif, 19000, Algérie.
3
Faculté des sciences et de la technologie, Université Mohamed Elbachir El Ibrahimi de Bordj Bou
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Arreridj, Bordj Bou Arreridj, 34000, Algérie.
Abstract
The pseudopotential plane-wave approach in the framework of the density functional theory, and
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the density functional perturbation theory with the generalized gradient approach for the exchange-correlation functional has been used to calculate the structural phase stability, elastic
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constants and thermodynamic properties of boron-arsenide (BAs) compound. The BAs compound
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transforms from the zincblende phase to rock-salt structure; the phase transition pressure was found to be 141.2 GPa with a volume contraction of around 8.2 %. The thermodynamic properties
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under high pressure and temperature up to 125 GPa and 1200 K respectively were also determined, analyzed and discussed in comparison with other data of the literature. The systematic
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errors in the static energy were corrected using the bpscal EEC method. Our results agree well with those reported in the literature, where for example, our calculated melting temperature (2116 K) deviates from the theoretical one (2132.83 K) with only 0.8 %, and the deviation between our result (1.86) of the Grüneisen parameter and the theoretical one (1.921) is only around 3.2 %.
* Corresponding author. E-mail: [email protected]
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Keywords: Phase transition; Thermodynamic properties; Boron arsenide; High-temperature; High-pressure. PACS: 62.50.-p; 64.70.K-; 61.66.Fn; 65.40.-b,
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1. Introduction Boron based pnictides compounds have started gaining attention in recent time due to their tremendous technological promise due to low ionicity, short bond length, high hardness, high melting point and wide band gap [1-6]. The BN compound appears to be more studied
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experimentally and theoretically compared to other ones. Experimentally, the structural phase transition, and the Birch coefficients of the equation of state of BAs compound were determined by Greene et al. [7]. Under high pressure, the low pressure phase is destabilized and structural phase transition occurs. For BAs compound, the phase transition from the zincblende (B3) phase
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to an amorphous state was detected; the limit pressure obtained of the stability of B3 phase is
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around 125 GPa [7]. Whereas the Birch coefficients of the equation of state are: 4.777 Å, 148 ± 6 GPa, and 3.9 ± 0.3 for the equilibrium lattice parameter a0, bulk modulus B0, and the pressure
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derivative of the bulk modulus B0′ respectively [7]. There are other works which reports the first principles calculations and some other methods on
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the study of BAs compound [8-14]. Labidi et al. [15] have performed a first principles calculation
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and reported pressure dependence of electronic and optical properties for BAs compound in B3 structure. They found that the effect of pressure on the main band gaps, show that the band gap min decreases
with increasing of pressure.
Furthermore, the scarce data on the thermodynamic properties under high pressure and temperature motivates us to perform a systematic study of these properties for BAs, which may shed some light on this material for its possible technological applications at high pressure and
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temperature. In the present work, we report on first-principles calculations of the structural parameters, the pressure of the phase transition and the temperature and pressure effect on the thermodynamic properties of BAs in B3 structure using the pseudopotential plane wave method in the framework of the density-functional theory (DFT) [16]. Besides, the elastic properties and
(DFPT) [16] were also obtained, examined and presented.
2. Computational methods
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some other derivative quantities in the framework of the density functional perturbation theory
In the present study, the calculations were carried out using ABINIT code [17], which is based on
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the pseudopotential plane wave (PP-PW) approach in the framework of DFT and DFPT. The interactions between the valence electrons and the nuclei and core electrons were described by Troullier-Martins type norm-conserving pseudopotentials, which have been generated thanks to
gradient approach (GGA) [19].
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the FHI98PP code [18]. The exchange-correlation energy was evaluated in the generalized
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A set of convergence tests was performed in order to correctly choose the mesh of k-points and the cutoff kinetic energy in the plane waves to start the ground state calculations. In order to find an
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appropriate energy cutoff, the total energy was calculated as a function of energy cutoff. The electronic wave functions are represented in a plane wave basis set with a kinetic energy cutoff of
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60 Ha. The Brillouin zone was sampled by (6x6x6) Monkhorst and Pack mesh of k points [20].
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We have also checked more dense grids but the results did not changed considerably. Convergence tests prove that the Brillouin zone sampling (6x6x6) and the kinetic energy cutoff 60 Ha were sufficient to guarantee a good convergence. In order to obtain the pressure and temperature dependence of the thermodynamic properties of BAs compound, the Debye model [21, 22] was successfully applied; our calculations are implemented through the Gibbs code [22]. The Gibbs code is used to obtain much thermodynamic quantities from the minimization of Energy-Volume (E, V) data, which is in our case obtained
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from the ABINIT code [17]. Gibbs implements several models for the inclusion of temperature effects to the results of an ab-initio calculation. The Debye model requires only the input of the static Energy-Volume (E, V) data and optionally the experimental Poisson’s ratio [21, 22]. So through the Debye model, one could calculate several thermodynamic quantities including the volume V, isothermal bulk modulus BT, etc. For more detail on the calculation of the
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thermodynamic properties of materials and the different relationships used here, please see the Refs. [21-24].
3. Results and discussion
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3.1. Equilibrium structural parameters
Usually, the structural equilibrium parameters can be predicted from Ab-initio calculation, using the pressure - unit cell volume (P-V) data, or the energy - unit cell volume (E-V) data. In order to obtain the structural parameters of BAs compound in both cubic zincblende (B3) and sodium
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chloride (B1) structures the energy - unit cell volume (E-V) data was used. The equilibrium lattice
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constant a0, bulk modulus B0 and pressure derivative of bulk modulus B0′ have been computed by minimizing the total energy by means of Murnaghan’s equation of state (EoS) [25] V / V B0 BV 1 '0 0 0' B0 1 B0 1 '
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BV E V E V0 0 ' B0
(1)
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Where V0 is the equilibrium volume, V is the volume, and E (V0) is the equilibrium energy.
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Fig. 1 presents the variation of the total energy as a function of the unit cell volume of BAs compound in both B3 and B1 structures. Our calculated values of a0, B0, and B0′ are summarized in Table 1 and compared with other theoretical results [1-6] and experimental data [7]. From the results summarized in Table 1, it is observed that our calculated values of a0 in both structures are in good agreement compared to available data [1-7]. It is important to note that the lattice parameter is one of the fundamental structural quantities for a solid directly related to bond length,
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which is determined by the density of valence electrons between the atoms of crystal. BAs compound is one of boron pnictides materials which in general have smaller unit cell volume and larger bulk modulus comparatively to some other conventional III-V compounds. For B3 structure, our obtained value (4.8342 Å) of a0 overestimate the experimental one (4.777 Å) [7] by only less than 1.2 %.
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Our calculated values of B0 and B0′ of both structures are generally also in agreement with the available theoretical data [1-6]. Where for example, our obtained value (126.74 GPa) of B0 of B3 structure underestimate the theoretical value (131 GPa) calculated by El Haj Hassan [3] by only
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about 3.25 %. 3.2. Pressure of the phase transition
The first method used to predict the threshold pressure of the phase transition is the dynamic calculation (phonon) at different pressures, which can find the dynamic instability (negative
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frequencies) and therefore the corresponding transition pressure. The second is from the
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thermodynamic calculation, the phase transition occurs when a change in the structure appears, which is caused by the variation of the free energy. The change in free energy becomes zero at
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phase transition pressure. However, the stability of any particular structure corresponds to the lowest Gibbs free energy, which is given by [8, 24]. (2)
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G = U + PV - TS
Where: U is the internal energy, P the pressure, T the temperature, S the entropy and V is the
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volume.
To investigate the pressure-induced structural transition of BAs, we have optimized at T = 0 K the cell parameters and atomic positions for both B3 and B1 phases. At T = 0 K, the Gibbs free energy becomes equal to the enthalpy [8, 24] H = U + PV
(3)
The computed enthalpies versus pressure curves for both phases of interest are shown in Fig. 2.
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From Fig. 2, we can see that the Gibbs’ free energy for B3 phase is more negative than this of B1 phase. The enthalpy versus pressure curves corresponding to B3 and B1 phases cross at a pressure of 141.2 GPa, suggesting therefore that the transition pressure (Pt) from B3 to B1 is 141.2 GPa. For comparison, our value obtained in the present work is listed in Table 2, along with the available experimental [7], and other theoretical ones [6, 8, 11-14, 26, 27].
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During the analysis of the phase transition from B3 to B1 phase, we also examined the reduction in the volume during this transition. The equation of state curves of BAs compound (plotted between V(P)/V(0) and pressure) for both B3 and B1 phases obtained in this work are plotted in Fig. 3. From these curves, one can estimate the volume collapse at the point of transition. The
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value of the volume collapse for BAs compound computed in this work is about 8.2 %. For comparison, our value obtained in the present work is listed also in Table 2 along with previous theoretical data [8, 11, 14, 26-28]. It is notable that our calculated value of the volume collapse is
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in good agreement compared to theoretical ones (9 %) obtained by Wentzcovitch and Cohen [14], (9.93 %) obtained in our previous work [8], and (7.92 %) obtained by Sarwan et al. [26]. Our
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obtained value (8.2 GPa) of the volume collapse overestimate the theoretical one (7.92 %) [26] by only about 3.54 %; so our result of the volume collapse is localized between these two theoretical
PT
ones. We note also that Cui et al. [11] found the phase transition B3 → B1 occurs at around 134
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0 %).
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GPa, but with no changing in the volume of the unit cell during this transition (Volume collapse =
3.3. Elastic properties The elastic properties of material define the mechanical behavior of solid that undergoes stress, deforms and then returns to its original shape after stress ceases [27]. The DFPT method was recently widely applied to calculate several physical properties, such as dynamic, piezoelectric and elastic properties of materials. The calculations of the elastic constants presented in this work were carried out using the ABINIT code [17] in the framework of the DFPT (For more detail on the
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calculation of the elastic constants with the DFPT method, please see for example our previous work [8]). For crystals with cubic structures, the matrix of the elastic constants contains only 3 independent elastic constants (C11, C12 and C44), the aggregate bulk modulus in this case is given as [8]: B (C11 2C12 ) / 3
(4)
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The evaluated elastic constants Cij, and the bulk modulus B of BAs at zero-pressure are presented in Table 3, along those of the literature [2, 6, 8, 10, 13, 26]. From the Table 3, we observe that our calculated elastic constants Cij are in general in reasonably good agreement with other theoretical ones [2, 6, 8, 10, 13, 26], where for example, the value (264.53 GPa) of C11 obtained by us is
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localized between the two theoretical values (275 GPa) and (252 GPa) obtained by Meradji et al. [2] and Thakore et al. [13] respectively. The value (62.88 GPa) of C12 obtained by us deviates from the theoretical one (63 GPa) reported by Meradji et al. [2] with only about 0.2 %.
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On the macroscopic scale, the elastic properties were usually described using the elastic moduli (Young modulus E, Poisson's ratio σ,…etc), which are varied to the direction of the single crystal.
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The Young's modulus E is a measure of the ability of solids to defend longitudinal stress [29]. For
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the aggregate polycrystalline material, the Young modulus E and Poisson's ratio σ are calculated with the following equations [29]:
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E 9B G/ (3B G) , and σ (3B 2G) / 2(3B G)
(5)
Where: B is the bulk modulus and G is the isotropic shear modulus.
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Our calculated values of G, E and σ are presented in Table 3, and compared with other available theoretical data of the literature [6, 10, 13, 26]. It is seen also that, our calculated values of G, E and σ are in general in good agreement with the previously calculated data [6, 10, 13, 26], where for example, the value (0.14) of σ obtained by us is localized between the two theoretical values (0.146) and (0.136) obtained in our previous work [8] and by Sarwan et al. [26] respectively. In addition our value of σ is equal exactly that reported by Bing et al. [6]. As we see in Table 3, BAs
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compound has a large value of G which implies a high level of resistant of this compound against the shear strain. The anisotropy factor A is quantity, which indicts the degree of the elastic anisotropy in material. For a completely isotropic material, the value of A is a unity (A = 1) [26, 29]. If the value of A differs than 1, so the solid shows the degree of elastic anisotropy. It can be calculated as follow
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[30, 31] : A 2C44 / (C11 C12 ) . The calculated value of A for BAs is around 1.43, it is greater than 1 which indicates that our material of interest shows elastic anisotropic character. Our obtained value of the anisotropy factor A is in good agreement compared to the values 1.445 and 1.461 reported by Wang and Ye [10] and Sarwan et al. [26]. The deviations between our value of A and
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the values reported by Wang and Ye [10] and Sarwan et al. [26] are around 1.04 % and 2.12 % respectively. If A <1, the crystal is stiffest along <100> cube axes, and when A >1, it is stiffest along the <111> body diagonals [30, 31].
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For cubic diamond and cubic zinc-blende crystals, the Kleinman internal strain parameter is quantity which describing the relative position of the cation and anion sub-lattices. In the central
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force model, this quantity can be expressed as function of the elastic constants C11 and C12 as
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follow [5] (C11 8C12 ) / (7C11 2C12 )
(6)
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It is known that a low value of ζ implies there is a large resistance against bond bending or bondangle distortion and vice versa [10]. Our obtained value of ζ of BAs is equal to 0.388; it is in
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excellent agreement with the theoretical value (0.39) reported by Ustundag et al. [5]. The deviation between our value and the value reported by Ustundag et al. [5] is only around 0.5 %. Our value of ζ is slightly higher than the theoretical value 0.362 reported by Wang and Ye [10]. For cubic polycrystalline materials, the average sound velocity vm is usually used to study the acoustic wave propagation; this quantity is given by the following expression [8, 26, 29]
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vm 1/3 (2/vt 3 ) (1/vl 3 )
10
1/3
(7)
Where: vl and vt are the longitudinal and transverse elastic wave velocities respectively, which are determined by using the following expressions [8, 29]: vl
3B 4G / 3 , and
vt G /
(8)
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where ρ is the crystal density. The calculated longitudinal (vl), transverse (vt) and average (vm) sound velocities of BAs
compound are: 7669 m/s, 4975 m/s and 5458 m/s respectively. They are presented in Table 4, and compared with other available theoretical data of the literature [6, 8, 13, 26, 27, 32]. Our results of
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vl, vt and vm are in general in very good agreement compared with our previous work [8] and with the results reported by Ustundag et al. [5], where for example the deviation between our value (5458 m/s) of vm and the value (5490 m/s) reported in our previous work [8] is only around 0.6 %.
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From Table 4, we can see that the values of vl, vt and vm obtained previously by Sarwan et al. [26] are very higher than our values and other data of the literature.
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One of the standard methods largely used to estimate the Debye temperature θD of solid is from the elastic constants and sound velocity data; it can be calculated from the average sound velocity
PT
vm using the following formula [8]:
(9)
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θD h/k B 3/4 Va 1/3vm
Where: h is the Planck constant; kB is the Boltzmann constant, and Va the atomic volume.
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The calculated Debye temperature θD of BAs at zero-pressure is about 672 K. It is listed in Table 4 and compared with available theoretical [5, 8, 26, 27] and experimental [32] data. It can observed that our calculated Debye temperature θD is slightly higher than the experimental one (625 K) [32], but in general it is in good agreement with other theoretical ones of the literature [5, 8], where for example, the deviation between our value (672 K) and the theoretical one (693.09 K) reported by Ustundag et al. [5] is only around 3 %.
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The melting temperature Tm (expressed in K) of several solids with cubic structure can be estimated from the elastic constant C11 (expressed in GPa) according to the following empirical relationship [8] Tm 553 5.91 C11
(10)
The calculated melting temperature Tm of BAs at zero-pressure is about 2116 K. It is listed in
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Table 4 and compared with available theoretical [5, 8] and experimental [32] data. It is seen that, our calculated melting temperature Tm is also in good agreement with other data of the literature [5, 8, 32], where for example, the value (2116 K) obtained by us deviates from the theoretical one
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(2132.83 K ) reported by Ustundag et al. [5] with only about 0.8 %. 3.4. Thermodynamic properties
The thermodynamic properties of materials may play an important role in physics of solid state matter [23], the good knowledge of these properties can allow extending our knowledge on their
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specific behavior when undergoing severe constraints of high pressure and temperature [33]. It is
ED
found that the empirical energy corrections (EECs) are able to reproduce the thermodynamic properties and EoS of solid materials at very high temperature [34]. So, in order to improve the
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systematic deviation of PBE functional used here, and to reproduce the experimental results in the range of the validity of the quasi-harmonic Debye model approximation, the empirical energy
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correction (EEC) becomes necessary. One of the methods used to correct the systematic errors in
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the static energy Esta (V) is that called bpscal EEC [34]. The BPSCAL correction modifies the static energy according to [34] ~
Esta (V ) Esta
B exp Vexp (V0 ) B0 V0
Esta
VV 0 V exp
Esta
(V0 )
(11)
Where (V0, B0) are calculated static equilibrium properties, and (Vexp, Bexp) are experimental ones. The experimental values of the volume Vexp and the bulk modulus Bexp used here to correct the total energy are measured by Greene et al. [7]; they are equal to 183 (a.u)3 and 148 GPa
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respectively. In the present work, the thermal properties are determined in the temperature range from 0 K to 1200 K, and the pressure effect is studied in the range of 0-125 GPa. Figures 4 (a) and 4 (b) display the pressure dependence of V at various temperatures and the temperature dependence of the V at various pressures for BAs compound. Note that for our compound of interest, V decreases
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with increasing pressure at all temperatures of interest. At any given pressure, V increases with raising temperature. The change in the behavior of the volume with pressure is obvious as applied pressure brings the atoms closer and hence it reduces the lattice parameter which in turn leads to the decrease in the volume [23]. On the other hand, when the temperature increases, this leads the
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dilatation of the lattice parameter leading thus to the increase in the volume of our crystal of interest. The similar trend was also observed in the case of BSb semiconducting compound [24]. The bulk modulus is one of the most important mechanical parameters of material; it usually
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decreases as temperature rises. The isothermal bulk modulus BT is given by the following expression [21-23],
ED
BT = -V(dp/dV)T = V(d 2 F/dV 2 )T
(12)
PT
where F is the vibrational Helmholtz free energy. Figure 5 displays the pressure and temperature dependence of BT for BAs compound. We observe
CE
that BT increases with increasing pressure at all considered temperatures. The behavior is
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monotonic for BAs. It should also be noted that for each given pressure BT decreases with raising temperature up to 1200 K. One may thus conclude that the material under study becomes harder with increasing pressure or decreasing temperature. At zero pressure and zero temperature, the obtained value of BT of BAs compound is 149.5 GPa, which is in excellent agreement with the experimental one (148 ± 6 GPa) reported by Greene et al. [7]. Our obtained value of BT overestimates the experimental data (148 GPa) by only around 1 %. At zero pressure, BT increases almost quadratically with increasing pressure; where at 1200 K, it
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becomes equal to 118.5 GPa. The behavior of BT under pressure and temperature is very similar to that found in BSb material [24]. The knowledge of the heat capacity of a material not only provides essential information on its vibrational properties but also is mandatory for many technological applications [35]. The heat capacity at constant volume Cv is obtained using the expression [23, 24]
where U is the total internal energy, and T is the temperature.
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(13)
CV = (dU/dT)V
Figures 6 (a) and 6 (b) show the pressure dependence of Cv at various temperatures and the temperature dependence of Cv at various pressures for BAs compound. It can be seen from Fig. 6
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(b) that Cv increases exponentially with the temperature for T < 500 K. For temperatures higher than 500 K, it follows naturally the Debye model and approaches the DuLong-Petit limit indicating that at high temperature all phonon modes are excited by the thermal energy [23]. It can
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be noted that similar qualitative behavior has been reported for the heat capacity Cv versus temperature for BSb semiconducting compound [24], magnesium silicate (MgSiO3) perovskite
ED
material [35], superconducting MgCNi3 material [36], and for both wurtzite and zinc-blende GaN
PT
compound [37]. At zero-pressure and 300 K, the calculated Cv of BAs is around 42 J.mol-1.K-1; it is very higher than the heat capacities of wurtzite and zinc-blende GaN compound, which are 34.4
CE
and 34.3 J.mol-1.K-1 respectively [37]. The heat capacity at constant pressure Cp can be obtained using the expression: CP (H / T) P
AC
[24], where: H is the enthalpy. Figure 7 shows the calculated Cp for different pressures and temperatures for BAs compound. It can be seen from Fig. 7 that the heat capacity Cp increases exponentially with the temperature for T < 500 K, and becomes almost stable (especially at very high pressure) for T ˃ 500 K. The similar qualitative behavior has been reported for Cp versus temperature for magnesium silicate (MgSiO3) perovskite [35].
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The effect of temperature and pressure on the entropy S is presented in Fig. 8. At low temperature and zero-pressure, one can note that the entropy increases quickly with raising the temperature T, then the behavior of the entropy as a function of temperature becomes more like a sub-linear behavior. By increasing pressure from 0 up to 125 GPa, the behavior of the entropy versus temperature remains qualitatively almost the same. However, from the quantitative point of view,
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the entropy seems to decrease slightly in magnitude with increasing pressure for BAs. This is in agreement with the variation of S with pressure and temperature deduced for BSb compound [24]. Figures 9 (a) and (b) show the pressure dependence of the Debye temperature θD at various temperatures and the temperature dependence of θD at various pressures for BAs compound. It can
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be seen from Fig. 9 that the Debye temperature increases monotonically with increasing pressure. At a fixed pressure, the Debye temperature decreases with raising temperature for BAs compound. At zero pressure and zero temperature, the obtained value of θD for BAs compound is around
M
570.5 K. This value is slightly lower than the experimental one (625 K) reported in Ref. [32]. Almost our obtained value of θD is slightly lower than the measured value (625 K) [32], it can
ED
observed that, our calculated value of θD is better than the theoretical ones (511.53 K) and (329.19 K) reported by Sarwan et al. [26] and Varshney et al. [27] respectively.
PT
In the quasi-harmonic Debye model approximation [21, 22], the Grüneisen parameter can be
CE
given by the following expression [21-23],
d ln D (V ) / d ln V
(14)
AC
Where V is the volume and D (V ) is the Debye temperature. Figure 10 displays the pressure and temperature dependence of the Grüneisen parameter of BAs compound. As can be seen from Fig. 10, the Grüneisen parameter for BAs decreases with increasing pressure and increases with temperature. At zero pressure and ambient temperature, our obtained value of the Grüneisen parameter is around 1.86. This value is in good agreement
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15
compared to the theoretical one (1.921) reported by Varshney et al. [27]. The deviation between our value (1.86) and the theoretical one (1.921) reported in Ref. [27] is only around 3.2 %. The thermal expansion coefficient α is a quantity which is used for describing the temperature effect on the size of object (dimensions change with a change in temperature) [38]. The volumetric thermal expansion coefficient α and the Grüneisen parameter are related as follow [23]: γ CV /BTV
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(15)
In Figure 11, we plot α as a function of temperature and pressure for BAs compound. The
behavior of this quantity is similar to that observed for the constant volume heat capacity and the
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entropy. At zero-pressure and T = 300 K, our obtained value of α is around 3.27 x 10-5 K-1. Our calculations show that α increases monotonically with increasing temperature and decreases with increasing pressure. This behavior can be explained that the anharmonic effect becomes less important at high pressures. With the increase of pressure, the volume of the solid reduces and the
M
atoms come closer to each other, increasing the depth of the potential energy well and reducing the anharmonic nature of the potential energy curve at high temperatures [35]. It can note that a
ED
similar qualitative behavior has been reported for α versus temperature for BSb [24]
PT
semiconducting compound, and for magnesium silicate (MgSiO3) perovskite material [35]. Since thermal expansivity of material is a result of anharmonicity in the potential energy; this quantity
AC
[35].
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becomes less dependent of temperature and pressure in the high temperature and pressure domain
4. Conclusion In summary, we have investigated the EoS parameters, the phase transition pressure, the elastic properties, and many other thermodynamical variables of BAs in B3 structure using density functional theory. We have calculated the ground state properties like lattice parameter, bulk modulus and first order pressure derivative of bulk modulus which are found in good agreement
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with the earlier available results. Our static calculations for BAs reveal that this compound follows B3-B1 phase transition. The phase transition pressure from B3 to B1 phase of the BAs compound has been determined using the total energy, and the Gibbs free energy. Our obtained value of 141.2 GPa is in general in agreement with other theoretical data. The unit cell volume, the isothermal bulk modulus, the heat capacity, the Debye temperature, the
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Grüneisen parameter and some other thermodynamic properties under high pressure up to 125 GPa and temperature up to 1200 K were also determined, analyzed and discussed. Our results agree well with those reported in the literature, where for example, the deviation between our result (1.86) of the Grüneisen and the theoretical one (1.921) is only around 3.2 %. At high
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temperatures, CV converges to a near-constant value (DuLong-Petit limit), while S, CP and θD increase monotonously with the temperature. All thermodynamic properties our compound of interest were found vary monotonically with either temperature or pressure.
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Figure Captions:
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Figure 1: Total energy versus volume of BAs compound in both B3 and B1 structures.
PT
-258
-261 -262
CE
-260
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Energy (eV)
-259
Zinc-blende phase Rock-salt phase
-263 -264
16
20
24
28
32 3
Volume (Å )
36
ACCEPTED MANUSCRIPT Figure 2: Enthalpies versus pressure for both B3 and B1 structures of BAs. -240
B1 B3
-250
-255
Pt = 141.2 GPa
-260
-265 0
30
60
90
120
150
180
M
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Pressure (GPa)
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Entalpy (eV)
-245
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Figure 3: Variations of the relative volume (Vp/V0) of B3 and B1-BAs versus pressure. 1.00
PT
0.95 0.90
0.75 0.70
(B3)
CE
0.80
AC
Vp/V0
0.85
0.65 0.60
(Vp(B3)-Vp(B1))/V0 = 8.2 %
(B1)
0.55 0.50
0
40
80
120
Pressure (GPa)
160
200
19
ACCEPTED MANUSCRIPT Figure 4: (a) Volume versus pressure at different temperatures of BAs compound, (b) Volume versus temperature at different pressures of BAs compound.
T= 0 K T= 300 K T= 800 K T= 1200 K
(a)
28
3
Volume (Å )
26
22 20 18
0
25
50
75
100
125
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Pressure (GPa) 28.5
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
27.0
(b)
3
Volume (Å )
25.5
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24.0 22.5
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21.0
18.0
0
PT
19.5
300
600
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Temperature (K)
AC
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24
900
1200
20
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Figure 5: (a) Isothermal bulk modulus BT versus pressure at different temperatures of BAs compound, (b) BT versus temperature at different pressures of BAs compound.
540
T= 0 K T= 300 K T= 800 K T= 1200 K
480
(a)
360 300 240 180 120 0
25
50
75
100
125
AN US
Pressure (GPa)
540 480
360
240
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
M
(b)
300
ED
BT (GPa)
420
180
PT
120 60
CR IP T
BT (GPa)
420
0
300
600
AC
CE
Temperature (K)
900
1200
ACCEPTED MANUSCRIPT Figure 6: (a) Constant volume heat capacity CV as a function of pressure at various temperatures of BAs compound, (b) CV as a function of temperature at various pressures of BAs. 50 48 46
42 40
T= 300 K T= 600 K T= 800 K T= 1200 K
(a)
38 36 34 32 30 0
25
50
75
100
125
AN US
Pressure (GPa) 50
30
20
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
M
(b)
ED
Cv (J/mol K)
40
PT
10
0
CR IP T
Cv (J/mol K)
44
0
300
600
AC
CE
Temperature (K)
900
1200
22
ACCEPTED MANUSCRIPT
23
Figure 7: (a) Constant pressure heat capacity CP versus pressure at various temperatures of BAs compound, (b) CP versus temperature at various pressures of BAs. 56 52
44 40
T= 300 K T= 600 K T= 800 K T= 1200 K
(a)
36 32 28
0
25
50
75
100
125
AN US
Pressure (GPa) 56 48
M
32 24
(b)
16
ED
Cp (J/mol K)
40
PT
8 0
CR IP T
Cp (J/mol K)
48
0
300
600
AC
CE
Temperature (K)
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
900
1200
ACCEPTED MANUSCRIPT Figure 8: (a) Entropy versus pressure at different temperatures of BAs compound, (b) Entropy versus temperature at different pressures of BAs compound. 105
(a)
T= 300 K T= 600 K
T= 800 K T= 1200 K
75 60 45 30 15
0
25
50
75
100
125
AN US
Pressure (GPa)
105 90
M
60 45 30
ED
S (J/mol K)
75
(b)
PT
15 0
CR IP T
S (J/mol K)
90
0
300
600
AC
CE
Temperature (K)
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
900
1200
24
ACCEPTED MANUSCRIPT Figure 9: (a) Debye temperature as a function of pressure at various temperatures of BAs compound, (b) Debye temperature as a function of temperature at various pressures of BAs.
935 850
T= 0 K T= 300 K T= 800 K T= 1200 K
(a)
765 680 595 510
0
25
50
75
100
125
AN US
Pressure (GPa)
935 850
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
M
765
ED
680 595 510
PT
Debye temperature (K)
1020
(b)
0
300
600
CE
Temperature (K)
AC
CR IP T
Debye temperature (K)
1020
900
1200
25
ACCEPTED MANUSCRIPT Figure 10: (a) Grüneisen parameter versus pressure at different temperatures of BAs compound, (b) Grüneisen parameter versus temperature at different pressures of BAs compound.
T= 0 K T= 300 K T= 800 K T= 1200 K
(a)
1.89 1.80 1.71 1.62 1.53 1.44 1.35 1.26 0
25
50
75
100
125
AN US
Pressure (GPa)
1.89
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
1.80
(b)
M
1.62 1.53
ED
Grüneisen parameter
1.98
1.71
1.44 1.35
PT
1.26 0
300
600
CE
Temperature (K)
AC
CR IP T
Grüneisen parameter
1.98
900
1200
26
ACCEPTED MANUSCRIPT
27
Figure 11: (a) Volumetric thermal expansion coefficient versus pressure at different temperatures of BAs compound, (b) Volumetric thermal expansion coefficient versus temperature at different pressures of BAs.
T= 300 K T= 600 K T= 800 K T= 1200 K
(a)
4,50
3,00 2,25 1,50 0,75 0
25
50
75
100
4,8 4,0
ED
-5
(10 /K)
2,4 1,6
PT
0,8 0,0
P = 0 GPa P = 25 GPa P = 75 GPa P = 125 GPa
M
(b)
3,2
125
AN US
Pressure (GPa)
0
300
600
AC
CE
Temperature (K)
CR IP T
-5
(10 /K)
3,75
900
1200
ACCEPTED MANUSCRIPT
28
Table Captions: TABLE 1: Equilibrium lattice parameter, bulk modulus, and its pressure derivative of BAs compound in both cubic zincblende and cubic sodium chloride structures.
B1
Table 1:
Other Works
parameter
work
a0 (Å)
4.8342
4.728 a, 4.812 b, 4.743 c, 4.814d, 4.784 e, 4.812 f , 4.779g, 4.777
B0 (GPa)
126.74
144 a, 133 b, 152c, 131d, 137 e, 130.91f , 138.29 g, 148 ± 6 h
B'0
3.58
4.00 a, 3.75 b, 3.65c, 3.49 e, 3.71 f , 4.09 g, 3.9 ± 0.3 h
a0 (Å)
4.6168
4.611 b, 4.534 c, 4.619 e, 4.622 f , 4.581 g
B0 (GPa)
114.56
132 b, 158c, 135e, 125.18f , 142.88 g
B'0
3.88
3.76 b, 3.55c, 3.44 e, 2.976f , 3.825 g
h
, a Ref. [1], b Ref. [2] for GGA calculation, c Ref. [2] for LDA calculation, d Ref. [3], e Ref. [4], f Ref. [5],
Ref. [7] Exp.
g
Ref. [6],
ED
M
h
Present
CR IP T
B3
EoS
AN US
Phase
TABLE 2: Transition pressure from B3 to B1 structure, and the corresponding volume contraction for
Parameter Pt (GPa)
CE
PT
BAs semiconducting compound.
Other Works
141.2
113.42 a, 125 b, 125 c, 134 d, 98 e, 110 f, 110 g , 93 h, 110 i
8.2
9.93c , 0 d, 9 g , 7.92 h, 4.2 i, 15 j
AC
ΔV/V0 (%)
Present work
Table 2: a Ref. [6], b Ref. [7] Exp,
c
Ref. [8],
d
Ref. [11], e Ref. [12], f Ref. [13],
g
Ref. [14], h Ref. [26], i Ref. [27] , j Ref. [28].
ACCEPTED MANUSCRIPT
29
TABLE 3: Elastic constants of BAs semiconducting compound in cubic zincblende structure.
Present work
Other Works
C11 (GPa)
264.53
275 a, 295 b, 286.39 c, 301.26 d, 291.4 e, 252 f, 289 g
C12 (GPa)
62.88
63 a, 78 b, 70.96c, 77.23 d, 72.8 e, 78 f, 70 g
C44 (GPa)
143.89
150 a, 177 b, 57.50 c, 163.87 d, 157.9 e, 128 f, 160 g
B (GPa)
130.10
133 a, 152 b, 142.77 c, 151.91 d, 138 e, 148 f, 143 g
G (GPa)
124.78
135.26 c,
E (GPa)
283.65
308.38 c, 262.3 e, 312 g
σ
0.14
0.14 c, 0.146 d, 0.2 e, 0.136 g c
Ref. [6], d Ref. [8], e Ref. [10], f Ref. [13], g Ref. [26].
M
AN US
Table 3: a Ref. [2] for GGA calculation, b Ref. [2] for LDA calculation,
CR IP T
Parameter
This work
(m/s)
7669
(m/s)
4975
vt
(m/s) vm
Tm (K)
7779.69 a 7373b, 15816.03 c, 8350 d 5114.69 a 5039 b , 8188 c, 4500 d
5458
5601.24a 5490 b, 10884.2c, 6023 d,
672
693.09 a, 698.16 b, 511.53c, 329.19 d, 625 e
AC
θD (K)
CE
vl
Others
PT
Parameter
ED
TABLE 4: Sound velocity, Debye temperature and melting temperature of BAs compound.
2116
2132.83 a, 2333.5 b, 2300 e
Table 4: a Ref. [5], b Ref. [8], c Ref. [26], d Ref. [27],
e
Ref. [32] Exp.