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Theoretical investigations on phase stability, elastic constants and electronic structures of Ga3Zr polymorphs under high pressure
Journal of Alloys and Compounds 696 (2017) 1010e1018
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Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Theoretical investigations on phase stability, elastic constants and electronic structures of Ga3Zr polymorphs under high pressure WenQing Ma a, *, Jing Zhang b, c, ** a
School of Material Science and Engineering, Xi'an Shiyou University, Xi'an 710065, Shaanxi, China School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, China c School of Computer Science and Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 June 2016 Received in revised form 26 October 2016 Accepted 2 December 2016 Available online 5 December 2016
By using density functional theory (DFT) method, the phase stability, elastic properties, Debye temperature and electronic structure of zirconium trigallide (Ga3Zr) have been investigated. The polymorphs of D022- and Ll2-Ga3Zr are compared under various pressures from 0 GPa to 60 GPa. The results show that phase transition from D022-Ga3Zr to Ll2-Ga3Zr will occur when pressure is over 28.48e36.34 GPa. With the pressure going up, the elastic constants, mechanical moduli and Debye temperature of both phases increase. For L12-Ga3Zr, better ductility and lower Debye temperature of L12-Ga3Zr is confirmed. The pressure-induced Zr-4d delocalization can strengthen its orbital hybridization with Ga(s,p), which leads to stronger atomic bonding, and subsequently makes the L12-Ga3Zr more stable under high pressure. © 2016 Published by Elsevier B.V.
Keywords: Density functional theory (DFT) Zirconium trigallide (Ga3Zr) Phase stability Elastic constants High pressure
1. Introduction The structure and properties of material can be affected by its surrounding conditions. Pressure is usually taken into account as a main factor. In recent years, by using density functional theory (DFT) method, many theoretical studies about pressure induced phase transition have been performed [1e4]. For GaeZr intermetallics, such as Ga3Zr and Ga2Zr, the crystallochemical affinity and optical functions [5], and their band structure, density of states, and crystal chemistry [6] were studied by using all-electron full potential linearized augmented plane wave (FP-LAPW) method. Furthermore, the phase diagram of GaeZr system were predicted by combining experimental investigation and thermodynamic modeling, the formation enthalpies of GaeZr intermetallics (including Ga3Zr, Ga2Zr, etc.) were calculated with DFT calculations, and the results agreed well with the available experimental data [7]. Under ambient conditions, the compound Ga3Zr exhibit tetragonal structure (D022, I4/mmm) [7,8], and the cubic structure (L12, Pm3m) is a metastable phase. When the pressure goes up,
Ga3Zr may transform from an original structure into another structure. However, according to our knowledge, the influence of high pressure on the properties of Ga3Zr is rarely reported. The present work is conducted to clarify the phase stability, elastic constants and electronic properties of D022- and L12-Ga3Zr under high pressures up to 60 GPa. 2. Methodology and details All calculations were performed with CASTEP [9,10] code, and plane-wave ultrasoft pseudopotentials [11,12] were employed. The exchange correlation (XC) energy was treated with two different forms: generalized gradient approximation (GGA) functionals of PBE [13], and local density approximation (LDA) functional of CAPZ [14,15]. The convergence threshold in the self consistent field (SCF) procedure was set as 5.0 107 eV/atom. In the minimization with BroydeneFletchereGoldfarbeShanno (BFGS) algorithm, the convergence tolerances for the energy, force, stress and displacement were set as 5.0 106 eV/atom, 0.01 eV/Å, 0.02 GPa and 5 104 Å, respectively. After the validation of convergence tests,
* Corresponding author. School of Materials Science and Engineering, Xi'an Shiyou University, Xi'an 710065, Shaanxi, China. ** Corresponding author. School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, China. E-mail addresses: [email protected] (W. Ma), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.jallcom.2016.12.030 0925-8388/© 2016 Published by Elsevier B.V.
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
the cutoff energy was set as 350 eV, the k-points grids were set as 20 12 20, 40 40 22, 24 24 10 and 24 24 24 for bulk a-Ga (A11, Cmca) [16], hcp-Zr (A3, P63/mmc) [17], D022- and L12Ga3Zr respectively. The above k-points settings ensured the separation of reciprocal space around 0.01 Å1. In order to ascertain the stability of D022- and L12-Ga3Zr phases, the formation energies of the both phases were calculated under high pressures up to 60 GPa. For the compound AmBn, the formation energy per atom (DEf) was ascertained as [18]:
DEf ðAm Bn Þ ¼
1 ½E ðAm Bn Þ mEbulk ðAÞ nEbulk ðBÞ; m þ n total (1)
where Etotal denotes the total energy per formula for the compound AmBn, and Ebulk denotes the total energy per atom of bulk A and B. The elastic constants of solids can be used to calculate their mechanical and thermodynamic properties. Commonly, the single crystal's elastic constants Cij can be obtained by calculating the total energy as a function of appropriate strains [19e21]. To calculate the elastic constants, a serial of deformed cells (strain) are introduced and optimized to calculate the tensor of elastic constants. The elastic strain energy U of a deformed crystal cell is given as [22]:
U¼
DE V0
¼
1X C ee 2 ij ij i j
(2)
where DE is the energy difference of deformed cell relative to the unstrained cell, V0 is the volume of the equilibrium cell without any deformation, Cijs are the elastic constants, ei and ej are strain. For the L12 structure, there are three independent elastic constants: C11 ¼ C22 ¼ C33, C12 ¼ C13 ¼ C23, C44 ¼ C55 ¼ C66; for the D022 structure, there are six independent constants: C11 ¼ C22, C12, C13 ¼ C23, C33, C44 ¼ C55, C66. The obtained monocystal quantities,
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such as these elastic constants, can not accurately stand for the properties of polycrystalline materials, so they should be further calculated and rectified. The polycrystalline mechanical quantities, such as bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (n), can be calculated from these independent elastic constants. There are three different algorithms corresponding to different bound to calculate these polycrystalline mechanical quantities: the Voigt bound is obtained by the average polycrystalline modules based on an assumption of uniform strain throughout a polycrystal, and it is the upper limit of the actual effective modules; while the Reuss bound is obtained by assuming a uniform stress, and it is the lower limit of the actual effective modules; the arithmetic average of Voigt and Reuss bounds is termed as the Voigt-Reuss-Hill approximation. The formula for calculating these mechanical quantities can be given as follows: (1) for the cubic L12 structures, BV ¼ BR ¼ (C11 þ 2C12)/3, GV ¼ (C11-C12þ3C44)/5, GR ¼ 5(C11-C12)C44/ [4C44þ3(C11-C12)] [23], (2) for the tetragonal D022 structures, BV¼(1/9)[2(C11þC12)þC33þ4C13], BR¼C2/M, GV¼(1/30)(Mþ3C113C12þ12C44þ6C66), GR ¼ 15{(18BV/C2)þ[6/(C11-C12)]þ(6/C44)þ(3/ C66)}1, M ¼ C11þC12þ2C33-4C13, C2¼(C11þC12)C33-2(C13)2 [24], and (3) for the Voigt-Reuss-Hill approximation, BH¼(1/2)(BV þ BR), GH¼(1/2)(GV þ GR), the sub letter of V, R and H denote Voigt, Reuss bound and Hill approximation, respectively. The BH and GH are adopted in this paper to calculate Young's modulus E and Poisson's ratio n by the following formulas: E ¼ 9BG/(3B þ G), n¼(3B-2G)/ [2(3B þ G)] [25].
3. Results and discussion 3.1. Methodology validation and pressure influences In order to validate the calculation methodology, the models of
a-Ga, hcp-Zr, D022-Ga3Zr and L12-Ga3Zr unit cells are fully
Table 1 Crystallographic data, lattice constants of bulk Ga, Zr and Ga3Zr. Phase
Space group (#)
Pearson symbol
Strukturbericht designation
Lattice parameters (Å)
a-Ga
Cmca (64)
oC8
A11
hcp-Zr
P63/mmc (194)
hP2
A3
Tetragonal Ga3Zr
I4/mmm (139)
tI8
D022
Cubic Ga3Zr
Pm3m(221)
cP4
L12
a a a a a a a a a
Calculation results ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
Experimental results
4.5776, b ¼ 7.7406, c ¼ 4.5727a 4.4177, b ¼ 7.4829, c ¼ 4.4271b 3.2298, c ¼ 5.1711a 3.1468, c ¼ 5.0793b 3.9160, c ¼ 9.1713a 3.8220, c ¼ 8.9522b 3.963, c ¼ 8.730c 4.1245a 4.0218b
a ¼ 4.519, b ¼ 7.658, c ¼ 4.526 [16] a ¼ 3.2331, c ¼ 5.1480 [17] a ¼ 3.971, c ¼ 8.729 [8]
Note. a Our calculated results with GGA-PBE potential. b Our calculated results with LDA-CAPZ potential. c Previous calculation results of D022-Ga3Zr in Ref. [7].
Table 2 Elastic constants (Cij) and formation energy (DE f) of L12- and D022-Ga3Zr.
DE f (Ga3Zr) (kJ/mol)
Phases
Data source
Elastic properties (GPa) C11
C12
C13
C33
C44
C66
B0
D022- Ga3Zr
Present calculation with GGA-PBE Present calculation with LDA-CAPZ Other calculation with GGA-PBE in Ref. [7] Present calculation with GGA-PBE Present calculation with LDA-CAPZ
181.16 219.37 e 133.30 149.19
85.14 105.39 e 78.06 95.33
47.26 59.36 e e e
180.68 215.49 e e e
75.79 88.98 e 42.06 50.32
113.76 135.32 e e e
99.89 121.96 e 96.47 113.28
L12- Ga3Zr
Note. a An approximate value of first-principle calculation with GGA-PBE potential under 0 GPa in Ref. [7].
46.304 52.606 53.727a 43.923 50.433
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W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
Table 3 Lattice parameters (a, c, a/c), volume per formula (V) and formation energy (DEf) of D022- and L12-Ga3Zr under various pressures. Phases Parameters
D022
L12
a (Å)
XC potentials Pressure (GPa)
GGA-PBE LDA-CAPZ c (Å) GGA-PBE LDA-CAPZ c/a GGA-PBE LDA-CAPZ V (Å3/formula) GGA-PBE LDA-CAPZ DEf (kJ/mol) GGA-PBE LDA-CAPZ a (Å) GGA-PBE LDA-CAPZ V (Å3/formula) GGA-PBE LDA-CAPZ DEf (kJ/mol) GGA-PBE LDA-CAPZ
0
5
10
15
20
25
30
35
40
45
50
55
60
3.9160 3.8220 9.1713 8.9522 2.3420 2.3423 70.3212 65.3855 46.304 52.606 4.1245 4.0218 70.1639 65.0521 43.923 50.433
3.8613 3.7775 9.0126 8.8186 2.3341 2.3345 67.1873 62.9185 5.520 4.232 4.0597 3.9691 66.9086 62.5282 7.747 2.343
3.8164 3.7395 8.8806 8.7076 2.3270 2.3285 64.6726 60.8829 55.225 42.424 4.0062 3.9233 64.2981 60.3885 57.210 43.989
3.7781 3.7063 8.7706 8.6098 2.3214 2.3230 62.5959 59.1349 103.194 87.666 3.9608 3.8841 62.1368 58.5964 104.878 88.831
3.7439 3.6769 8.6734 8.5248 2.3167 2.3185 60.7866 57.6259 149.738 131.678 3.9213 3.8501 60.2962 57.0711 151.034 132.426
3.7139 3.6504 8.5885 8.4483 2.3125 2.3143 59.2308 56.2886 194.973 174.626 3.8861 3.8187 58.6870 55.6861 195.928 174.934
3.6866 3.6263 8.5120 8.3790 2.3089 2.3106 57.8434 55.0921 239.096 216.593 3.8548 3.7910 57.2803 54.4830 239.639 216.458
3.6619 3.6038 8.4421 8.3157 2.3054 2.3075 56.6022 53.9996 282.224 257.704 3.8266 3.7654 56.0324 53.3867 282.343 257.114
3.6392 3.5832 8.3769 8.2569 2.3019 2.3043 55.4709 53.0065 324.468 298.030 3.8002 3.7419 54.8807 52.3934 324.144 296.974
3.6179 3.5639 8.3188 8.2025 2.2993 2.3016 54.4432 52.0915 365.885 337.634 3.7764 3.7198 53.8560 51.4705 365.120 336.112
3.5981 3.5460 8.2630 8.1518 2.2965 2.2989 53.4877 51.2508 406.554 376.576 3.7537 3.6992 52.8906 50.6202 405.341 374.585
3.5797 3.5289 8.2118 8.1040 2.2940 2.2965 52.6140 50.4601 446.533 414.904 3.7324 3.6799 51.9954 49.8320 444.895 412.438
3.5622 3.5132 8.1640 8.0598 2.2918 2.2941 51.7976 49.7393 485.875 452.664 3.7125 3.6619 51.1681 49.1043 483.772 449.715
Note: For the calculations of formation energy, the reference states are a-Ga and hcp-Zr under zero pressure and zero temperature.
optimized under 0 GPa. The calculated lattice parameters (a and c) are very close to the available experimental data (presented in Table 1). Furthermore, the elastic constants of D022- and L12-Ga3Zr under zero pressure are calculated with GGA-PBE and LDA-CAPZ
Fig. 1. Pressure dependence of (a) volume ratio (V/V0) and (b) lattice parameter ratios (a/a0, c/c0) (note: a0, c0 and V0 denote the parameters under zero pressure).
functionals, and the results are presented in Table 2. The comparing of elastic constants between our results and other literature are unavailable, due to the previous data of Ga3Zr are rarely reported. However, the bulk modulus of Al3Zr at 0 GPa was empirically predicted as B0 ¼ 95.4 GPa [26], and the elastic constants of L12-Al3Zr were calculated as C11 ¼ 184.8 GPa, C12 ¼ 59.9 GPa, C44 ¼ 72.0 GPa, B0 ¼ 101.5 GPa [27] with GGA-PW91 potential [28]. Comparing with these data of Al3Zr, our calculation results of Ga3Zr are accordant and reasonable. The formation energy DEf (Ga3Zr) are also calculated under 0 GPa, our results of D022-Ga3Zr are 46.304 kJ/mol and 52.606 kJ/mol calculated with GGA and LDA potentials, respectively. The values are close to the data (53.727 kJ/mol) in previous literature [7]. Summarily, the adequate accuracy can be guaranteed by using the above calculation parameters and XC functionals. Then, the unit cell models of D022- and L12-Ga3Zr have been fully optimized under various high pressures up to 60 GPa, the lattice parameters (a, c, c/a), volume per formula (V) and formation energy [DEf(Ga3Zr)] are summarized in Table 3. Comparing with the lattice parameters, volume and formation energies calculated with LDA-CAPZ potential, the results obtained with GGA-PBE are slightly
Fig. 2. Calculated formation energy difference [DEf(L12-Ga3Zr)eDEf(D022-Ga3Zr)] as a function of pressure, the critical pressures corresponding to the structural transition are labeled by arrows.
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
larger, which is accordant with the “over-binding” effect of LDA potential reported in other calculations [29e33], and moreover, the difference always exists from 0 GPa to 60 GPa, which suggests the numerical errors strongly depends on the exchange-correlation functional [34]. In order to clarify pressure influences on the structures of both phases, the pressure dependences of lattice parameter ratios (a/a0, c/c0) and volume ratio (V/V0) (a0, c0 and V0 denote the parameters under zero pressure) are illustrated in Fig. 1. By comparing V/V0pressure curves (Fig. 1-a), one can find that L12-Ga3Zr is more easily compressed than D022-Ga3Zr, and for the both phases, larger compressibility (or smaller values of V/V0) can be obtained with GGA-PBE than LDA-CAPZ. More important, when pressure goes up, the ratio c/c0 of D022Ga3Zr decreases more quickly than its ratio a/a0 (Fig. 1-b), this
Fig. 3. Calculated volume-pressure relationship for D022- and L12-Ga3Zr.
1013
phenomenon indicates that D022-Ga3Zr is more easily compressed along z direction. This can be interpreted as: the GaeZr bonds mainly locate in x-y plane (namely, (0 0 1) plane of D022-Ga3Zr unit cell), therefore, direction z of D022-Ga3Zr unit cell (lattice parameter c) is comparatively easier to be compressed than directions x and y (lattice parameter a). Besides that, lattice parameter ratio (a/a0) of L12-Ga3Zr approximately equals to the average value of ratios (a/a0, c/c0) of D022-Ga3Zr. 3.2. Formation energy and phase stability Generally, the chemical compound with smaller (or more negative) formation energy suggests it is more stable in thermodynamics. The elementary substances under zero pressure are adopted as reference states in our formation energy calculations. On the basis of a-Ga and hcp-Zr under 0 GPa, the formation energies of D022- and L12-Ga3Zr under various pressures are estimated according above equation, and the results are presented in Table 3. Under zero pressure, the formation energies of D022-Ga3Zr are 46.304 kJ/mol and 52.606 kJ/mol calculated with GGA and LDA potentials, respectively. The values of L12-Ga3Zr are 43.923 kJ/mol and 50.433 kJ/mol. Therefore, D022-Ga3Zr is more stable than L12-Ga3Zr under normal pressure. With pressure increasing, the formation energies of D022- and L12-Ga3Zr rapidly change toward positive side. More importantly, the difference [DEf(L12-Ga3Zr)eDEf(D022-Ga3Zr)] of both structures gradually decrease with the pressure going up. Therefore, it is reasonable to speculate that: although the D022-Ga3Zr is more stable than L12-Ga3Zr at normal pressure, the priority of D022 structure gradually becomes dimmer when pressure increases. Thus, the structural transition from D022 to L12 can be anticipated when pressure goes higher enough. The curves between formation energy difference, volume and pressure are shown in Figs. 2 and 3, respectively. In Fig. 2, the values of [DEf(L12-Ga3Zr) e DEf(D022-Ga3Zr)] are
Table 4 Elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E (all in GPa) and Poisson's ratio (n) of D022- and L12-Ga3Zr under various pressures. Phases
XC
Pressure (GPa)
C11
C12
C13
C33
C44
C66
B
G
E
n
B/G
D022
GGA
0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60
181.16 243.22 291.45 348.75 393.63 440.77 483.77 219.37 277.84 329.51 379.69 428.34 475.23 520.62 133.30 174.39 224.87 282.63 330.22 365.04 409.99 149.19 197.38 255.69 313.31 356.32 394.90 435.20
85.14 125.93 164.27 201.94 241.88 278.78 312.47 105.39 143.65 183.85 222.48 259.96 296.46 332.07 78.06 118.48 152.03 182.98 214.98 247.41 280.78 95.33 135.00 168.39 200.50 234.42 268.83 303.04
47.26 74.85 104.49 131.46 158.80 186.45 210.87 59.36 86.96 116.33 145.14 173.00 200.28 226.93 e e e e e e e e e e e e e e
180.68 249.60 307.84 363.54 419.49 477.09 524.41 215.49 233.02 341.89 397.12 451.38 503.13 555.06 e e e e e e e e e e e e e e
75.79 99.47 120.03 138.70 155.78 171.81 186.81 88.98 111.48 131.56 149.61 166.54 182.46 197.43 42.06 59.73 82.24 103.97 122.31 137.82 152.77 50.32 71.18 95.20 115.70 133.04 148.30 162.27
113.76 150.51 182.46 211.38 238.15 263.42 287.27 135.32 170.66 201.93 230.33 256.86 281.77 305.24 e e e e e e e e e e e e e e
99.89 142.64 181.60 220.74 257.98 295.40 328.52 121.96 156.08 203.31 241.96 279.49 315.87 351.49 96.47 137.12 176.31 216.20 253.39 286.62 323.85 113.28 155.79 197.49 238.10 275.05 310.85 347.09
74.47 96.24 112.60 130.05 143.41 157.23 169.80 87.97 104.94 125.34 140.52 154.91 168.35 181.15 35.53 44.04 59.31 77.39 90.42 97.93 108.16 39.15 51.12 69.62 86.72 97.26 105.21 113.17
178.93 235.72 279.94 326.10 362.97 400.61 434.52 212.76 257.18 311.92 353.19 392.25 428.87 463.77 94.95 119.34 159.99 207.42 242.42 263.76 291.99 105.33 138.23 186.89 231.99 261.01 283.64 306.24
0.20 0.22 0.24 0.25 0.27 0.27 0.28 0.21 0.23 0.24 0.26 0.27 0.27 0.28 0.34 0.35 0.35 0.34 0.34 0.35 0.35 0.35 0.35 0.34 0.34 0.34 0.35 0.35
1.34 1.48 1.61 1.70 1.80 1.88 1.93 1.39 1.49 1.62 1.72 1.80 1.88 1.94 2.71 3.11 2.97 2.79 2.80 2.93 2.99 2.89 3.05 2.84 2.75 2.83 2.95 3.07
LDA
L12
GGA
LDA
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W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
positive when the pressure less than 28.48 GPa or 36.34 GPa with GGA-PBE or LDA-CAPZ functionals. Therefore, D022-Ga3Zr is more stable when pressure less than 28.48e36.34 GPa. It is consistent with the previous argument that, in the (Al, Ga)e(Ti, Zr, Hf) systems, the D022 structure is more stable than L12 structure [35]. D022-Ga3Zr will transform into L12 structure when the pressure is higher than 28.48e36.34 GPa, and the volume will drop about DV/ V ¼ 1.0e1.1% as shown in Fig. 3. 3.3. Elastic constants and thermodynamic properties For the both phases, their elastic constants and mechanical moduli under high pressures (5e60 GPa) are also calculated with GGA-PBE and LDA-CAPZ potentials, and the results are presented in Table 4. The calculated elastic constant and moduli increase while pressure goes up. For D022-Ga3Zr, the elastic constants C11 and C33 characterize the incompressibility along x and z directions, respectively. It is confirmed that, under same pressure, the calculated C33 are larger than C11 of D022-Ga3Zr, which indicates that, if under uniaxial stresses, D022-Ga3Zr is more incompressible along z axis (c direction) than x axis (a direction). As for L12-Ga3Zr, the compressibility is denoted with C11 along both x and z directions. It is confirmed that L12-Ga3Zr will be more difficult to be compressed under higher pressure. For cubic L12-Ga3Zr, its independent elastic constants should satisfy the mechanical stability criteria as follows: C11 > 0, C12 > 0, (C11-C12)>0, C44 > 0 [36]. Our calculated results of L12-Ga3Zr under 0e60 GPa are in line with the criteria, and the mechanical stability of L12-Ga3Zr is confirmed up to 60 GPa. The requirements of mechanical stability in a tetragonal crystal are: C11 > 0, C12 > 0, C13 > 0, C33 > 0, C44 > 0, C66 > 0, (C11-C12)>0, (C11þC33-2C13)>0, and (2C11þC33þ2C12-4C13)>0 [36]. According to Table 4, even though D022-Ga3Zr will transform into L12-Ga3Zr under pressures higher than 28.48e36.34 GPa, the both structures satisfy the above criteria of mechanical stability under various pressures from 0 GPa to 60 GPa. For the independent elastic constants of D022-Ga3Zr and L12Ga3Zr obtained with GGA-PBE potential, the ratios of Cij/Cij0 are calculated, in which Cij and Cij0 denote the values calculate under high pressure and 0 GPa, respectively. Similarly, the ratios B/B0, G/ G0 and E/E0 are also calculated to ascertain the influence of pressure on elastic and mechanical properties. Their curves are illustrated in Fig. 4. For D022-Ga3Zr, the strengthening effect of Cij can be queued as C13 > C12 > C33 > C11 > C66 > C44; while for L12-Ga3Zr, the sequence is C44 > C12 > C11. Furthermore, for the mechanical moduli of D022- and L12-Ga3Zr, the strengthening effect can be queued as B > E > G with pressure increasing. The hardness and brittleness of the compounds also have a relation to the ratio (B/G) between bulk modulus and shear modulus: according to Pugh's criterion [37], the compound with larger B/G ratio (>~1.75) usually is ductile, and with smaller B/G ratio (<~1.75) usually is brittle. The results of ratio B/G are illustrated in Fig. 5. From Table 4 and Fig. 5, one can find that, the ratio B/ G of D022- and L12-Ga3Zr will be larger with pressure going up, it indicates that better ductility can be anticipated for the both structures of Ga3Zr under high pressures. However, for D022-Ga3Zr, the ratio of B/G constantly increases with the pressure going up. Meanwhile, for L12-Ga3Zr, the vast increasement of B/G is followed by a decreasing in pressure range of 10e30 GPa, and then a gradual increasing from 30 GPa to 60 GPa. Besides that, the B/G values of L12-Ga3Zr are larger than 1.75, while the values of D022-Ga3Zr are much lower, it can be interpreted that L12-Ga3Zr will exhibit better ductility than D022-Ga3Zr in the pressure range of 0e60 GPa. Furthermore, for D022-Ga3Zr, the B/G values is lower than 1.75 in
Fig. 4. Pressure dependences of elastic constants (Cij) and mechanical moduli (B, G, E) for D022- and L12-Ga3Zr (Cij0 and B0, G0, E0 are values calculated under 0 GPa).
Fig. 5. Raito B/G of D022- and L12-Ga3Zr under various pressures (the horizontal dash line denote the value of B/G ¼ 1.75).
the pressure range from 0 GPa to ~35 GPa, while larger than 1.75 in the range from ~35 GPa to 60 GPa. It suggests that D022-Ga3Zr will change from brittle to ductile when pressure is higher than ~35 GPa.
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
Besides the ratio of B/G, Poisson's ratio (n) is also commonly used to evaluate the materials' mechanical response under external stress, namely, larger Poisson's ratio usually corresponds to good ductility [38]. The value of n for covalent materials is small (n ¼ 0.1), and for ionic materials a typical value is 0.25. For central force solids, the value of n is between 0.25 and 0.5 [39]. In Table 4, the calculated Poisson's ratio of D022-Ga3Zr increases from 0.20 to 0.28, and for L12-Ga3Zr, it is roughly stable around 0.34e0.35 while pressure goes up from 0 GPa to 60 GPa. These results indicate that, for D022-Ga3Zr, ionic features exist in its interatomic interactions; and for L12-Ga3Zr, the phase is basically central force solid. Besides that, larger Poisson's ratio of L12-Ga3Zr indicates that Ga3Zr in L12 structure has better ductility than it in D022 structure. And in terms of ductility, D022-Ga3Zr exhibits more pressure sensitivity than L12Ga3Zr, which is in accordance with the analysis of Pugh's ratio (B/G). Debye temperature (QD) is a fundamental parameter for the materials' thermodynamic properties, and it is correlated with many physical properties such as specific heat, elastic constants, melting temperature, etc. The experimental value of a solid usually can be calculated from the sound velocity [40]. In this paper, Debye temperatures of D022- and L12-Ga3Zr under ambient and high pressure are estimated with the elastic constant data. QD may be estimated from the averaged sound velocity by the following equation [40]:
QD ¼
1 h 3n NA r0 3 ym k 4p M
(3)
1015
pressure increasing. 3.4. Electronic structure and atomic bonding For a deeper insight into the atomic bonding and electronic structure of Ga3Zr under high pressure, the valence electron density difference and partial density of states (PDOS) are investigated with GGA-PBE functional. Fig. 6 illustrates the valence electron density difference along (110) planes of D022- and L12-Ga3Zr under 0 GPa and 60 GPa, respectively. For the both structures, their partial density of states (PDOS) curves under zero and high (60 GPa) pressures are presented in Fig. 7. The electron density difference (Fig. 6) shows that, for the both phases, there are charge accumulation and dissipation around zirconium and gallium atoms, respectively. This case not only exists under zero pressure, but also high pressure (at 60 GPa). So, the charge transfer from Ga to Zr exists in the both phases, which is independent to the pressure. For the gallium atoms in D022-Ga3Zr unit cell, there are two different fractional coordinates: (0, 0, 0.5) and (0, 0.5, 0.25), and they are labeled as Ga1 and Ga2 for simplicity (shown as Fig. 6-a). From Fig. 6-a and b, one can find the charge accumulation mainly exists along the <112> directions of D022-Ga3Zr. And in local regions around Zr atoms, the charge accumulation mainly locates along <110> direction, namely between Zr and Ga1. Therefore, ZreGa1 bond has more covalent features, and larger bonding strength comparing with ZreGa2. From Fig. 6-c and d, one can find
where h denotes the Planck's constant, k denotes Boltzmann's constant, NA denotes the Avogadro constant, n denotes the atoms number per molecule, M denotes the molecular weight, and r0 denotes the density, respectively. The average sound velocity ym can be calculated as below:
"
ym ¼
1 2 1 þ 3 y3s y3 l
!#1=3 ;
(4)
and
yl ¼
1=2 3B þ 4G 1=2 G ; ys ¼ 3r0 r0
(5)
where yl and ys are the longitudinal and shear sound velocities, respectively. The Debye temperatures of D022- and L12-Ga3Zr are calculated from the aforementioned values of the lattice parameters, the bulk modulus and shear modulus, and the results are listed in Table 5. According to our knowledge, there rarely has any other report about Debye temperature of Ga3Zr, the comparison between our results with other data is unavailable. In Debye theory, the QD is the temperature of a crystal's highest normal mode of vibration, i.e., the highest temperature that can be achieved due to a single normal vibration. One can find that D022-Ga3Zr has higher QD than L12Ga3Zr, and their Debye temperatures will increase with the
Fig. 6. Valence charge density difference (e/Å3) of D022- and L12-Ga3Zr under 0 GPa and 60 GPa along (110) plane: (a) D022 0 GPa; (b) D022 60 GPa; (c) L12 0 GPa; (d) L12 60 GPa.
Table 5 Debye temperature QD (unit in K) of D022- and L12-Ga3Zr under various pressures. Phases
XC potential
Pressure(GPa) 0
10
20
30
40
50
60
D022-Ga3Zr
GGA-PBE LDA-CAPZ GGA-PBE LDA-CAPZ
409.62 440.24 287.48 298.34
460.40 475.97 316.21 337.02
493.92 516.57 362.75 389.11
527.07 543.63 410.38 430.64
550.43 567.77 440.45 453.38
573.41 589.12 455.93 469.19
593.13 608.51 476.72 484.55
L12-Ga3Zr
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W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
Fig. 7. PDOS of D022-Ga3Zr (a, b) and L12-Ga3Zr (c, d) under 0 GPa (a, c) and 60 GPa (b, d).
the bonding charges of L12-Ga3Zr mainly locate at tetrahedral interstices along <111> directions. More importantly, for the both phases, the electrons transfer between GaeZr atom pairs become stronger under 60 GPa (Fig. 6-b and d) comparing with they are at zero pressure (Fig. 4-a and c). This phenomenon indicates that the interatomic electron interaction of D022- and L12-Ga3Zr is significantly influenced by external pressure, and electron interaction will be more active in the circumstance with higher pressure. Therefore, stronger GaeZr bonds can be anticipated under high pressure. By using GGA-PBE potential, the partial density of states (PDOS) plots under 0 GPa and 60 GPa are calculated (shown as Fig. 7), and the k-points of 32 32 14 and 32 32 32 are adopted for D022and L12-Ga3Zr, respectively. In the PDOS curves, the states at Fermi level are not zero, and band gap does not exist, which indicates the both phases are conductors. Furthermore, comparing with D022Ga3Zr, the states above Fermi level of L12-Ga3Zr are more significant, which suggests L12-Ga3Zr has better electrical conductivity.
For the both phases, their DOS are mainly contributed from Zr4d, Ga-4p and Ga-4s. In the plot of D022-Ga3Zr under 0 GPa (Fig. 7a), the covalent bonds between Zr and Ga atoms, especially ZreGa1 atom pairs are confirmed by the resonant peaks around 1.2 eV, 5.2 eV, 6.8 eV. These bonds mainly derive from the electron orbitals hybridization of Zr-4d and Ga(s,p) states. Similarly, in the plot of L12-Ga3Zr under 0 GPa (Fig. 7-c), the resonant peaks exist around 2.5 eV and 6.3 eV, and for the electron states below Fermi level, the electrons' localization is much insignificant than D022-Ga3Zr under 0 GPa, which suggests the covalent bonds of L12-Ga3Zr is weaker than D022-Ga3Zr. By comparing PDOS plots of D022-Ga3Zr under 0 GPa and 60 GPa (Fig. 7-a and b), one can easily find that high pressure will make the curves' energy range widened towards both negative and positive sides. Especially for the electrons states higher than Fermi level, namely for “free electrons”, high pressure will lead to lower peaks, and around Fermi level, the curves' “valley” tends to be wider and shallower. Similar case exists in the plots of L12-Ga3Zr under 0 GPa
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
and 60 GPa (Fig. 7-c and d). High pressure will result in a wider bandwidth, especially wider conduction band. For D022-Ga3Zr under 0 GPa (Fig. 7-a), the energy range covered under PDOS curves is about 10.5 eVe6 eV; while for D022-Ga3Zr under 60 GPa (Fig. 7-b), the energy range is widened as about 12.5 eVe8 eV. For L12-Ga3Zr under 0 GPa (Fig. 7-c), the energy range is 11.2 eVe12 eV; while for L12-Ga3Zr under 60 GPa (Fig. 7-d), the energy range is widened as about 14 eVe14 eV. Consequently, for the both phases under 60 GPa, the wider conduction bands can be confirmed, which indicates that: for both D022- and L12-Ga3Zr, there are more metallic features under 60 GPa than them under 0 GPa, and better electrical conductivity under high pressure can be anticipated. For the valence band in PDOS plots of both phases, the highest curve's peak will shift towards negative energy level. For example, from 0 GPa to 60 GPa, the peaks of D022-Ga3Zr and L12-Ga3Zr will move from 1.2 eV and 2.5 eV to 1.8 eV and 3.8 eV, respectively. This phenomenon demonstrates that the covalent bonds of Ga3Zr will form in deeper energy level under high pressure, and bring about stronger bond strength. Summarily, it is reasonable to conclude that pressure-induced delocalization of Zr electrons, especially Zr-4d, which strengthens the electron hybridization and atomic bonding between Zr and Ga atoms. Consequently, it improves the phase stability of L12-Ga3Zr, even though the phase exhibits more metallic features with the pressure increasing. 4. Summary By using DFT calculation method, the atomic structure, formation energy, elastic constants and electronic structure of D022- and L12-Ga3Zr have been investigated with respect to external pressure ranging from 0 GPa to 60 GPa. The phase stability, mechanical moduli, Debye temperature and electronic interaction are discussed. The following results are obtained. (1) Pressure induced transition from D022-Ga3Zr to L12-Ga3Zr will occur when pressure is higher than 28.48e36.34 GPa, and volume drop is about 1%. (2) The elastic constants, mechanical moduli and Debye temperature of both phases increase with pressure going up. For D022-Ga3Zr, the strengthening effect of Cij can be queued as C13 > C12 > C33 > C11 > C66 > C44; while for L12-Ga3Zr, the sequence is C44 > C12 > C11; for the mechanical moduli of both phases, the sequence is B > E > G. L12-Ga3Zr has better ductility and lower Debye temperature than D022 structure at various pressures in the range of 0e60 GPa. (3) The charge transfer from Ga to Zr exists in the both phases, and electron interaction will be more active in the circumstance with higher pressure. The Zr-4d delocalization of Ll2Ga3Zr under high pressure will strengthen the hybridization between Zr-4d and Ga(s,p), which leads to stronger atomic bonding, and subsequently makes the L12-Ga3Zr more stable under high pressure. Acknowledgements The authors acknowledge the technical support from Center for High Performance Computing of Northwestern Polytechnical University. References [1] Y. Yao, D.D. Klug, B4-B1 phase transition of GaN under isotropic and uniaxial compression, Phys. Rev. B 88 (2013) 14113.
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Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigenvalue formalism, Phys. Rev. B 41 (1990) 7892e7895. [13] J.P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77 (1996) 3865. [14] J.P. Perdew, A. Zunger, Self-interaction correction to density-functional approximations for many-electron systems, Phys. Rev. B 23 (1981) 5048. [15] D.M. Ceperley, B.J. Alder, Ground state of the electron gas by a stochastic method, Phys. Rev. Lett. 45 (1980) 566. [16] H.G. von Schnering, R. Nesper, Alpha-gallium: an alternative to the boron structure, Acta Chem. Scand. 45 (1991) 870e872. [17] J. Goldak, L.T. Lloyd, C.S. Barrett, Lattice parameters, thermal expansions, and Grüneisen coefficients of zirconium, 4.2 to 1130 K, Phys. Rev. 144 (1966) 478e484. [18] G. Ghosh, First-principles calculations of structural energetics of CueTM (TM¼Ti, Zr, Hf) intermetallics, Acta Mater. 55 (2007) 3347e3374. [19] G. Ghosh, S. Vaynman, M. Asta, M.E. 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Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: http://www.elsevier.com/locate/jalcom
Theoretical investigations on phase stability, elastic constants and electronic structures of Ga3Zr polymorphs under high pressure WenQing Ma a, *, Jing Zhang b, c, ** a
School of Material Science and Engineering, Xi'an Shiyou University, Xi'an 710065, Shaanxi, China School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, China c School of Computer Science and Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, China b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 27 June 2016 Received in revised form 26 October 2016 Accepted 2 December 2016 Available online 5 December 2016
By using density functional theory (DFT) method, the phase stability, elastic properties, Debye temperature and electronic structure of zirconium trigallide (Ga3Zr) have been investigated. The polymorphs of D022- and Ll2-Ga3Zr are compared under various pressures from 0 GPa to 60 GPa. The results show that phase transition from D022-Ga3Zr to Ll2-Ga3Zr will occur when pressure is over 28.48e36.34 GPa. With the pressure going up, the elastic constants, mechanical moduli and Debye temperature of both phases increase. For L12-Ga3Zr, better ductility and lower Debye temperature of L12-Ga3Zr is confirmed. The pressure-induced Zr-4d delocalization can strengthen its orbital hybridization with Ga(s,p), which leads to stronger atomic bonding, and subsequently makes the L12-Ga3Zr more stable under high pressure. © 2016 Published by Elsevier B.V.
Keywords: Density functional theory (DFT) Zirconium trigallide (Ga3Zr) Phase stability Elastic constants High pressure
1. Introduction The structure and properties of material can be affected by its surrounding conditions. Pressure is usually taken into account as a main factor. In recent years, by using density functional theory (DFT) method, many theoretical studies about pressure induced phase transition have been performed [1e4]. For GaeZr intermetallics, such as Ga3Zr and Ga2Zr, the crystallochemical affinity and optical functions [5], and their band structure, density of states, and crystal chemistry [6] were studied by using all-electron full potential linearized augmented plane wave (FP-LAPW) method. Furthermore, the phase diagram of GaeZr system were predicted by combining experimental investigation and thermodynamic modeling, the formation enthalpies of GaeZr intermetallics (including Ga3Zr, Ga2Zr, etc.) were calculated with DFT calculations, and the results agreed well with the available experimental data [7]. Under ambient conditions, the compound Ga3Zr exhibit tetragonal structure (D022, I4/mmm) [7,8], and the cubic structure (L12, Pm3m) is a metastable phase. When the pressure goes up,
Ga3Zr may transform from an original structure into another structure. However, according to our knowledge, the influence of high pressure on the properties of Ga3Zr is rarely reported. The present work is conducted to clarify the phase stability, elastic constants and electronic properties of D022- and L12-Ga3Zr under high pressures up to 60 GPa. 2. Methodology and details All calculations were performed with CASTEP [9,10] code, and plane-wave ultrasoft pseudopotentials [11,12] were employed. The exchange correlation (XC) energy was treated with two different forms: generalized gradient approximation (GGA) functionals of PBE [13], and local density approximation (LDA) functional of CAPZ [14,15]. The convergence threshold in the self consistent field (SCF) procedure was set as 5.0 107 eV/atom. In the minimization with BroydeneFletchereGoldfarbeShanno (BFGS) algorithm, the convergence tolerances for the energy, force, stress and displacement were set as 5.0 106 eV/atom, 0.01 eV/Å, 0.02 GPa and 5 104 Å, respectively. After the validation of convergence tests,
* Corresponding author. School of Materials Science and Engineering, Xi'an Shiyou University, Xi'an 710065, Shaanxi, China. ** Corresponding author. School of Mechanical and Precision Instrument Engineering, Xi'an University of Technology, Xi'an 710048, Shaanxi, China. E-mail addresses: [email protected] (W. Ma), [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.jallcom.2016.12.030 0925-8388/© 2016 Published by Elsevier B.V.
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
the cutoff energy was set as 350 eV, the k-points grids were set as 20 12 20, 40 40 22, 24 24 10 and 24 24 24 for bulk a-Ga (A11, Cmca) [16], hcp-Zr (A3, P63/mmc) [17], D022- and L12Ga3Zr respectively. The above k-points settings ensured the separation of reciprocal space around 0.01 Å1. In order to ascertain the stability of D022- and L12-Ga3Zr phases, the formation energies of the both phases were calculated under high pressures up to 60 GPa. For the compound AmBn, the formation energy per atom (DEf) was ascertained as [18]:
DEf ðAm Bn Þ ¼
1 ½E ðAm Bn Þ mEbulk ðAÞ nEbulk ðBÞ; m þ n total (1)
where Etotal denotes the total energy per formula for the compound AmBn, and Ebulk denotes the total energy per atom of bulk A and B. The elastic constants of solids can be used to calculate their mechanical and thermodynamic properties. Commonly, the single crystal's elastic constants Cij can be obtained by calculating the total energy as a function of appropriate strains [19e21]. To calculate the elastic constants, a serial of deformed cells (strain) are introduced and optimized to calculate the tensor of elastic constants. The elastic strain energy U of a deformed crystal cell is given as [22]:
U¼
DE V0
¼
1X C ee 2 ij ij i j
(2)
where DE is the energy difference of deformed cell relative to the unstrained cell, V0 is the volume of the equilibrium cell without any deformation, Cijs are the elastic constants, ei and ej are strain. For the L12 structure, there are three independent elastic constants: C11 ¼ C22 ¼ C33, C12 ¼ C13 ¼ C23, C44 ¼ C55 ¼ C66; for the D022 structure, there are six independent constants: C11 ¼ C22, C12, C13 ¼ C23, C33, C44 ¼ C55, C66. The obtained monocystal quantities,
1011
such as these elastic constants, can not accurately stand for the properties of polycrystalline materials, so they should be further calculated and rectified. The polycrystalline mechanical quantities, such as bulk modulus (B), shear modulus (G), Young's modulus (E), Poisson's ratio (n), can be calculated from these independent elastic constants. There are three different algorithms corresponding to different bound to calculate these polycrystalline mechanical quantities: the Voigt bound is obtained by the average polycrystalline modules based on an assumption of uniform strain throughout a polycrystal, and it is the upper limit of the actual effective modules; while the Reuss bound is obtained by assuming a uniform stress, and it is the lower limit of the actual effective modules; the arithmetic average of Voigt and Reuss bounds is termed as the Voigt-Reuss-Hill approximation. The formula for calculating these mechanical quantities can be given as follows: (1) for the cubic L12 structures, BV ¼ BR ¼ (C11 þ 2C12)/3, GV ¼ (C11-C12þ3C44)/5, GR ¼ 5(C11-C12)C44/ [4C44þ3(C11-C12)] [23], (2) for the tetragonal D022 structures, BV¼(1/9)[2(C11þC12)þC33þ4C13], BR¼C2/M, GV¼(1/30)(Mþ3C113C12þ12C44þ6C66), GR ¼ 15{(18BV/C2)þ[6/(C11-C12)]þ(6/C44)þ(3/ C66)}1, M ¼ C11þC12þ2C33-4C13, C2¼(C11þC12)C33-2(C13)2 [24], and (3) for the Voigt-Reuss-Hill approximation, BH¼(1/2)(BV þ BR), GH¼(1/2)(GV þ GR), the sub letter of V, R and H denote Voigt, Reuss bound and Hill approximation, respectively. The BH and GH are adopted in this paper to calculate Young's modulus E and Poisson's ratio n by the following formulas: E ¼ 9BG/(3B þ G), n¼(3B-2G)/ [2(3B þ G)] [25].
3. Results and discussion 3.1. Methodology validation and pressure influences In order to validate the calculation methodology, the models of
a-Ga, hcp-Zr, D022-Ga3Zr and L12-Ga3Zr unit cells are fully
Table 1 Crystallographic data, lattice constants of bulk Ga, Zr and Ga3Zr. Phase
Space group (#)
Pearson symbol
Strukturbericht designation
Lattice parameters (Å)
a-Ga
Cmca (64)
oC8
A11
hcp-Zr
P63/mmc (194)
hP2
A3
Tetragonal Ga3Zr
I4/mmm (139)
tI8
D022
Cubic Ga3Zr
Pm3m(221)
cP4
L12
a a a a a a a a a
Calculation results ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼ ¼
Experimental results
4.5776, b ¼ 7.7406, c ¼ 4.5727a 4.4177, b ¼ 7.4829, c ¼ 4.4271b 3.2298, c ¼ 5.1711a 3.1468, c ¼ 5.0793b 3.9160, c ¼ 9.1713a 3.8220, c ¼ 8.9522b 3.963, c ¼ 8.730c 4.1245a 4.0218b
a ¼ 4.519, b ¼ 7.658, c ¼ 4.526 [16] a ¼ 3.2331, c ¼ 5.1480 [17] a ¼ 3.971, c ¼ 8.729 [8]
Note. a Our calculated results with GGA-PBE potential. b Our calculated results with LDA-CAPZ potential. c Previous calculation results of D022-Ga3Zr in Ref. [7].
Table 2 Elastic constants (Cij) and formation energy (DE f) of L12- and D022-Ga3Zr.
DE f (Ga3Zr) (kJ/mol)
Phases
Data source
Elastic properties (GPa) C11
C12
C13
C33
C44
C66
B0
D022- Ga3Zr
Present calculation with GGA-PBE Present calculation with LDA-CAPZ Other calculation with GGA-PBE in Ref. [7] Present calculation with GGA-PBE Present calculation with LDA-CAPZ
181.16 219.37 e 133.30 149.19
85.14 105.39 e 78.06 95.33
47.26 59.36 e e e
180.68 215.49 e e e
75.79 88.98 e 42.06 50.32
113.76 135.32 e e e
99.89 121.96 e 96.47 113.28
L12- Ga3Zr
Note. a An approximate value of first-principle calculation with GGA-PBE potential under 0 GPa in Ref. [7].
46.304 52.606 53.727a 43.923 50.433
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W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
Table 3 Lattice parameters (a, c, a/c), volume per formula (V) and formation energy (DEf) of D022- and L12-Ga3Zr under various pressures. Phases Parameters
D022
L12
a (Å)
XC potentials Pressure (GPa)
GGA-PBE LDA-CAPZ c (Å) GGA-PBE LDA-CAPZ c/a GGA-PBE LDA-CAPZ V (Å3/formula) GGA-PBE LDA-CAPZ DEf (kJ/mol) GGA-PBE LDA-CAPZ a (Å) GGA-PBE LDA-CAPZ V (Å3/formula) GGA-PBE LDA-CAPZ DEf (kJ/mol) GGA-PBE LDA-CAPZ
0
5
10
15
20
25
30
35
40
45
50
55
60
3.9160 3.8220 9.1713 8.9522 2.3420 2.3423 70.3212 65.3855 46.304 52.606 4.1245 4.0218 70.1639 65.0521 43.923 50.433
3.8613 3.7775 9.0126 8.8186 2.3341 2.3345 67.1873 62.9185 5.520 4.232 4.0597 3.9691 66.9086 62.5282 7.747 2.343
3.8164 3.7395 8.8806 8.7076 2.3270 2.3285 64.6726 60.8829 55.225 42.424 4.0062 3.9233 64.2981 60.3885 57.210 43.989
3.7781 3.7063 8.7706 8.6098 2.3214 2.3230 62.5959 59.1349 103.194 87.666 3.9608 3.8841 62.1368 58.5964 104.878 88.831
3.7439 3.6769 8.6734 8.5248 2.3167 2.3185 60.7866 57.6259 149.738 131.678 3.9213 3.8501 60.2962 57.0711 151.034 132.426
3.7139 3.6504 8.5885 8.4483 2.3125 2.3143 59.2308 56.2886 194.973 174.626 3.8861 3.8187 58.6870 55.6861 195.928 174.934
3.6866 3.6263 8.5120 8.3790 2.3089 2.3106 57.8434 55.0921 239.096 216.593 3.8548 3.7910 57.2803 54.4830 239.639 216.458
3.6619 3.6038 8.4421 8.3157 2.3054 2.3075 56.6022 53.9996 282.224 257.704 3.8266 3.7654 56.0324 53.3867 282.343 257.114
3.6392 3.5832 8.3769 8.2569 2.3019 2.3043 55.4709 53.0065 324.468 298.030 3.8002 3.7419 54.8807 52.3934 324.144 296.974
3.6179 3.5639 8.3188 8.2025 2.2993 2.3016 54.4432 52.0915 365.885 337.634 3.7764 3.7198 53.8560 51.4705 365.120 336.112
3.5981 3.5460 8.2630 8.1518 2.2965 2.2989 53.4877 51.2508 406.554 376.576 3.7537 3.6992 52.8906 50.6202 405.341 374.585
3.5797 3.5289 8.2118 8.1040 2.2940 2.2965 52.6140 50.4601 446.533 414.904 3.7324 3.6799 51.9954 49.8320 444.895 412.438
3.5622 3.5132 8.1640 8.0598 2.2918 2.2941 51.7976 49.7393 485.875 452.664 3.7125 3.6619 51.1681 49.1043 483.772 449.715
Note: For the calculations of formation energy, the reference states are a-Ga and hcp-Zr under zero pressure and zero temperature.
optimized under 0 GPa. The calculated lattice parameters (a and c) are very close to the available experimental data (presented in Table 1). Furthermore, the elastic constants of D022- and L12-Ga3Zr under zero pressure are calculated with GGA-PBE and LDA-CAPZ
Fig. 1. Pressure dependence of (a) volume ratio (V/V0) and (b) lattice parameter ratios (a/a0, c/c0) (note: a0, c0 and V0 denote the parameters under zero pressure).
functionals, and the results are presented in Table 2. The comparing of elastic constants between our results and other literature are unavailable, due to the previous data of Ga3Zr are rarely reported. However, the bulk modulus of Al3Zr at 0 GPa was empirically predicted as B0 ¼ 95.4 GPa [26], and the elastic constants of L12-Al3Zr were calculated as C11 ¼ 184.8 GPa, C12 ¼ 59.9 GPa, C44 ¼ 72.0 GPa, B0 ¼ 101.5 GPa [27] with GGA-PW91 potential [28]. Comparing with these data of Al3Zr, our calculation results of Ga3Zr are accordant and reasonable. The formation energy DEf (Ga3Zr) are also calculated under 0 GPa, our results of D022-Ga3Zr are 46.304 kJ/mol and 52.606 kJ/mol calculated with GGA and LDA potentials, respectively. The values are close to the data (53.727 kJ/mol) in previous literature [7]. Summarily, the adequate accuracy can be guaranteed by using the above calculation parameters and XC functionals. Then, the unit cell models of D022- and L12-Ga3Zr have been fully optimized under various high pressures up to 60 GPa, the lattice parameters (a, c, c/a), volume per formula (V) and formation energy [DEf(Ga3Zr)] are summarized in Table 3. Comparing with the lattice parameters, volume and formation energies calculated with LDA-CAPZ potential, the results obtained with GGA-PBE are slightly
Fig. 2. Calculated formation energy difference [DEf(L12-Ga3Zr)eDEf(D022-Ga3Zr)] as a function of pressure, the critical pressures corresponding to the structural transition are labeled by arrows.
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
larger, which is accordant with the “over-binding” effect of LDA potential reported in other calculations [29e33], and moreover, the difference always exists from 0 GPa to 60 GPa, which suggests the numerical errors strongly depends on the exchange-correlation functional [34]. In order to clarify pressure influences on the structures of both phases, the pressure dependences of lattice parameter ratios (a/a0, c/c0) and volume ratio (V/V0) (a0, c0 and V0 denote the parameters under zero pressure) are illustrated in Fig. 1. By comparing V/V0pressure curves (Fig. 1-a), one can find that L12-Ga3Zr is more easily compressed than D022-Ga3Zr, and for the both phases, larger compressibility (or smaller values of V/V0) can be obtained with GGA-PBE than LDA-CAPZ. More important, when pressure goes up, the ratio c/c0 of D022Ga3Zr decreases more quickly than its ratio a/a0 (Fig. 1-b), this
Fig. 3. Calculated volume-pressure relationship for D022- and L12-Ga3Zr.
1013
phenomenon indicates that D022-Ga3Zr is more easily compressed along z direction. This can be interpreted as: the GaeZr bonds mainly locate in x-y plane (namely, (0 0 1) plane of D022-Ga3Zr unit cell), therefore, direction z of D022-Ga3Zr unit cell (lattice parameter c) is comparatively easier to be compressed than directions x and y (lattice parameter a). Besides that, lattice parameter ratio (a/a0) of L12-Ga3Zr approximately equals to the average value of ratios (a/a0, c/c0) of D022-Ga3Zr. 3.2. Formation energy and phase stability Generally, the chemical compound with smaller (or more negative) formation energy suggests it is more stable in thermodynamics. The elementary substances under zero pressure are adopted as reference states in our formation energy calculations. On the basis of a-Ga and hcp-Zr under 0 GPa, the formation energies of D022- and L12-Ga3Zr under various pressures are estimated according above equation, and the results are presented in Table 3. Under zero pressure, the formation energies of D022-Ga3Zr are 46.304 kJ/mol and 52.606 kJ/mol calculated with GGA and LDA potentials, respectively. The values of L12-Ga3Zr are 43.923 kJ/mol and 50.433 kJ/mol. Therefore, D022-Ga3Zr is more stable than L12-Ga3Zr under normal pressure. With pressure increasing, the formation energies of D022- and L12-Ga3Zr rapidly change toward positive side. More importantly, the difference [DEf(L12-Ga3Zr)eDEf(D022-Ga3Zr)] of both structures gradually decrease with the pressure going up. Therefore, it is reasonable to speculate that: although the D022-Ga3Zr is more stable than L12-Ga3Zr at normal pressure, the priority of D022 structure gradually becomes dimmer when pressure increases. Thus, the structural transition from D022 to L12 can be anticipated when pressure goes higher enough. The curves between formation energy difference, volume and pressure are shown in Figs. 2 and 3, respectively. In Fig. 2, the values of [DEf(L12-Ga3Zr) e DEf(D022-Ga3Zr)] are
Table 4 Elastic constants Cij, bulk modulus B, shear modulus G, Young's modulus E (all in GPa) and Poisson's ratio (n) of D022- and L12-Ga3Zr under various pressures. Phases
XC
Pressure (GPa)
C11
C12
C13
C33
C44
C66
B
G
E
n
B/G
D022
GGA
0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60 0 10 20 30 40 50 60
181.16 243.22 291.45 348.75 393.63 440.77 483.77 219.37 277.84 329.51 379.69 428.34 475.23 520.62 133.30 174.39 224.87 282.63 330.22 365.04 409.99 149.19 197.38 255.69 313.31 356.32 394.90 435.20
85.14 125.93 164.27 201.94 241.88 278.78 312.47 105.39 143.65 183.85 222.48 259.96 296.46 332.07 78.06 118.48 152.03 182.98 214.98 247.41 280.78 95.33 135.00 168.39 200.50 234.42 268.83 303.04
47.26 74.85 104.49 131.46 158.80 186.45 210.87 59.36 86.96 116.33 145.14 173.00 200.28 226.93 e e e e e e e e e e e e e e
180.68 249.60 307.84 363.54 419.49 477.09 524.41 215.49 233.02 341.89 397.12 451.38 503.13 555.06 e e e e e e e e e e e e e e
75.79 99.47 120.03 138.70 155.78 171.81 186.81 88.98 111.48 131.56 149.61 166.54 182.46 197.43 42.06 59.73 82.24 103.97 122.31 137.82 152.77 50.32 71.18 95.20 115.70 133.04 148.30 162.27
113.76 150.51 182.46 211.38 238.15 263.42 287.27 135.32 170.66 201.93 230.33 256.86 281.77 305.24 e e e e e e e e e e e e e e
99.89 142.64 181.60 220.74 257.98 295.40 328.52 121.96 156.08 203.31 241.96 279.49 315.87 351.49 96.47 137.12 176.31 216.20 253.39 286.62 323.85 113.28 155.79 197.49 238.10 275.05 310.85 347.09
74.47 96.24 112.60 130.05 143.41 157.23 169.80 87.97 104.94 125.34 140.52 154.91 168.35 181.15 35.53 44.04 59.31 77.39 90.42 97.93 108.16 39.15 51.12 69.62 86.72 97.26 105.21 113.17
178.93 235.72 279.94 326.10 362.97 400.61 434.52 212.76 257.18 311.92 353.19 392.25 428.87 463.77 94.95 119.34 159.99 207.42 242.42 263.76 291.99 105.33 138.23 186.89 231.99 261.01 283.64 306.24
0.20 0.22 0.24 0.25 0.27 0.27 0.28 0.21 0.23 0.24 0.26 0.27 0.27 0.28 0.34 0.35 0.35 0.34 0.34 0.35 0.35 0.35 0.35 0.34 0.34 0.34 0.35 0.35
1.34 1.48 1.61 1.70 1.80 1.88 1.93 1.39 1.49 1.62 1.72 1.80 1.88 1.94 2.71 3.11 2.97 2.79 2.80 2.93 2.99 2.89 3.05 2.84 2.75 2.83 2.95 3.07
LDA
L12
GGA
LDA
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W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
positive when the pressure less than 28.48 GPa or 36.34 GPa with GGA-PBE or LDA-CAPZ functionals. Therefore, D022-Ga3Zr is more stable when pressure less than 28.48e36.34 GPa. It is consistent with the previous argument that, in the (Al, Ga)e(Ti, Zr, Hf) systems, the D022 structure is more stable than L12 structure [35]. D022-Ga3Zr will transform into L12 structure when the pressure is higher than 28.48e36.34 GPa, and the volume will drop about DV/ V ¼ 1.0e1.1% as shown in Fig. 3. 3.3. Elastic constants and thermodynamic properties For the both phases, their elastic constants and mechanical moduli under high pressures (5e60 GPa) are also calculated with GGA-PBE and LDA-CAPZ potentials, and the results are presented in Table 4. The calculated elastic constant and moduli increase while pressure goes up. For D022-Ga3Zr, the elastic constants C11 and C33 characterize the incompressibility along x and z directions, respectively. It is confirmed that, under same pressure, the calculated C33 are larger than C11 of D022-Ga3Zr, which indicates that, if under uniaxial stresses, D022-Ga3Zr is more incompressible along z axis (c direction) than x axis (a direction). As for L12-Ga3Zr, the compressibility is denoted with C11 along both x and z directions. It is confirmed that L12-Ga3Zr will be more difficult to be compressed under higher pressure. For cubic L12-Ga3Zr, its independent elastic constants should satisfy the mechanical stability criteria as follows: C11 > 0, C12 > 0, (C11-C12)>0, C44 > 0 [36]. Our calculated results of L12-Ga3Zr under 0e60 GPa are in line with the criteria, and the mechanical stability of L12-Ga3Zr is confirmed up to 60 GPa. The requirements of mechanical stability in a tetragonal crystal are: C11 > 0, C12 > 0, C13 > 0, C33 > 0, C44 > 0, C66 > 0, (C11-C12)>0, (C11þC33-2C13)>0, and (2C11þC33þ2C12-4C13)>0 [36]. According to Table 4, even though D022-Ga3Zr will transform into L12-Ga3Zr under pressures higher than 28.48e36.34 GPa, the both structures satisfy the above criteria of mechanical stability under various pressures from 0 GPa to 60 GPa. For the independent elastic constants of D022-Ga3Zr and L12Ga3Zr obtained with GGA-PBE potential, the ratios of Cij/Cij0 are calculated, in which Cij and Cij0 denote the values calculate under high pressure and 0 GPa, respectively. Similarly, the ratios B/B0, G/ G0 and E/E0 are also calculated to ascertain the influence of pressure on elastic and mechanical properties. Their curves are illustrated in Fig. 4. For D022-Ga3Zr, the strengthening effect of Cij can be queued as C13 > C12 > C33 > C11 > C66 > C44; while for L12-Ga3Zr, the sequence is C44 > C12 > C11. Furthermore, for the mechanical moduli of D022- and L12-Ga3Zr, the strengthening effect can be queued as B > E > G with pressure increasing. The hardness and brittleness of the compounds also have a relation to the ratio (B/G) between bulk modulus and shear modulus: according to Pugh's criterion [37], the compound with larger B/G ratio (>~1.75) usually is ductile, and with smaller B/G ratio (<~1.75) usually is brittle. The results of ratio B/G are illustrated in Fig. 5. From Table 4 and Fig. 5, one can find that, the ratio B/ G of D022- and L12-Ga3Zr will be larger with pressure going up, it indicates that better ductility can be anticipated for the both structures of Ga3Zr under high pressures. However, for D022-Ga3Zr, the ratio of B/G constantly increases with the pressure going up. Meanwhile, for L12-Ga3Zr, the vast increasement of B/G is followed by a decreasing in pressure range of 10e30 GPa, and then a gradual increasing from 30 GPa to 60 GPa. Besides that, the B/G values of L12-Ga3Zr are larger than 1.75, while the values of D022-Ga3Zr are much lower, it can be interpreted that L12-Ga3Zr will exhibit better ductility than D022-Ga3Zr in the pressure range of 0e60 GPa. Furthermore, for D022-Ga3Zr, the B/G values is lower than 1.75 in
Fig. 4. Pressure dependences of elastic constants (Cij) and mechanical moduli (B, G, E) for D022- and L12-Ga3Zr (Cij0 and B0, G0, E0 are values calculated under 0 GPa).
Fig. 5. Raito B/G of D022- and L12-Ga3Zr under various pressures (the horizontal dash line denote the value of B/G ¼ 1.75).
the pressure range from 0 GPa to ~35 GPa, while larger than 1.75 in the range from ~35 GPa to 60 GPa. It suggests that D022-Ga3Zr will change from brittle to ductile when pressure is higher than ~35 GPa.
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
Besides the ratio of B/G, Poisson's ratio (n) is also commonly used to evaluate the materials' mechanical response under external stress, namely, larger Poisson's ratio usually corresponds to good ductility [38]. The value of n for covalent materials is small (n ¼ 0.1), and for ionic materials a typical value is 0.25. For central force solids, the value of n is between 0.25 and 0.5 [39]. In Table 4, the calculated Poisson's ratio of D022-Ga3Zr increases from 0.20 to 0.28, and for L12-Ga3Zr, it is roughly stable around 0.34e0.35 while pressure goes up from 0 GPa to 60 GPa. These results indicate that, for D022-Ga3Zr, ionic features exist in its interatomic interactions; and for L12-Ga3Zr, the phase is basically central force solid. Besides that, larger Poisson's ratio of L12-Ga3Zr indicates that Ga3Zr in L12 structure has better ductility than it in D022 structure. And in terms of ductility, D022-Ga3Zr exhibits more pressure sensitivity than L12Ga3Zr, which is in accordance with the analysis of Pugh's ratio (B/G). Debye temperature (QD) is a fundamental parameter for the materials' thermodynamic properties, and it is correlated with many physical properties such as specific heat, elastic constants, melting temperature, etc. The experimental value of a solid usually can be calculated from the sound velocity [40]. In this paper, Debye temperatures of D022- and L12-Ga3Zr under ambient and high pressure are estimated with the elastic constant data. QD may be estimated from the averaged sound velocity by the following equation [40]:
QD ¼
1 h 3n NA r0 3 ym k 4p M
(3)
1015
pressure increasing. 3.4. Electronic structure and atomic bonding For a deeper insight into the atomic bonding and electronic structure of Ga3Zr under high pressure, the valence electron density difference and partial density of states (PDOS) are investigated with GGA-PBE functional. Fig. 6 illustrates the valence electron density difference along (110) planes of D022- and L12-Ga3Zr under 0 GPa and 60 GPa, respectively. For the both structures, their partial density of states (PDOS) curves under zero and high (60 GPa) pressures are presented in Fig. 7. The electron density difference (Fig. 6) shows that, for the both phases, there are charge accumulation and dissipation around zirconium and gallium atoms, respectively. This case not only exists under zero pressure, but also high pressure (at 60 GPa). So, the charge transfer from Ga to Zr exists in the both phases, which is independent to the pressure. For the gallium atoms in D022-Ga3Zr unit cell, there are two different fractional coordinates: (0, 0, 0.5) and (0, 0.5, 0.25), and they are labeled as Ga1 and Ga2 for simplicity (shown as Fig. 6-a). From Fig. 6-a and b, one can find the charge accumulation mainly exists along the <112> directions of D022-Ga3Zr. And in local regions around Zr atoms, the charge accumulation mainly locates along <110> direction, namely between Zr and Ga1. Therefore, ZreGa1 bond has more covalent features, and larger bonding strength comparing with ZreGa2. From Fig. 6-c and d, one can find
where h denotes the Planck's constant, k denotes Boltzmann's constant, NA denotes the Avogadro constant, n denotes the atoms number per molecule, M denotes the molecular weight, and r0 denotes the density, respectively. The average sound velocity ym can be calculated as below:
"
ym ¼
1 2 1 þ 3 y3s y3 l
!#1=3 ;
(4)
and
yl ¼
1=2 3B þ 4G 1=2 G ; ys ¼ 3r0 r0
(5)
where yl and ys are the longitudinal and shear sound velocities, respectively. The Debye temperatures of D022- and L12-Ga3Zr are calculated from the aforementioned values of the lattice parameters, the bulk modulus and shear modulus, and the results are listed in Table 5. According to our knowledge, there rarely has any other report about Debye temperature of Ga3Zr, the comparison between our results with other data is unavailable. In Debye theory, the QD is the temperature of a crystal's highest normal mode of vibration, i.e., the highest temperature that can be achieved due to a single normal vibration. One can find that D022-Ga3Zr has higher QD than L12Ga3Zr, and their Debye temperatures will increase with the
Fig. 6. Valence charge density difference (e/Å3) of D022- and L12-Ga3Zr under 0 GPa and 60 GPa along (110) plane: (a) D022 0 GPa; (b) D022 60 GPa; (c) L12 0 GPa; (d) L12 60 GPa.
Table 5 Debye temperature QD (unit in K) of D022- and L12-Ga3Zr under various pressures. Phases
XC potential
Pressure(GPa) 0
10
20
30
40
50
60
D022-Ga3Zr
GGA-PBE LDA-CAPZ GGA-PBE LDA-CAPZ
409.62 440.24 287.48 298.34
460.40 475.97 316.21 337.02
493.92 516.57 362.75 389.11
527.07 543.63 410.38 430.64
550.43 567.77 440.45 453.38
573.41 589.12 455.93 469.19
593.13 608.51 476.72 484.55
L12-Ga3Zr
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W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
Fig. 7. PDOS of D022-Ga3Zr (a, b) and L12-Ga3Zr (c, d) under 0 GPa (a, c) and 60 GPa (b, d).
the bonding charges of L12-Ga3Zr mainly locate at tetrahedral interstices along <111> directions. More importantly, for the both phases, the electrons transfer between GaeZr atom pairs become stronger under 60 GPa (Fig. 6-b and d) comparing with they are at zero pressure (Fig. 4-a and c). This phenomenon indicates that the interatomic electron interaction of D022- and L12-Ga3Zr is significantly influenced by external pressure, and electron interaction will be more active in the circumstance with higher pressure. Therefore, stronger GaeZr bonds can be anticipated under high pressure. By using GGA-PBE potential, the partial density of states (PDOS) plots under 0 GPa and 60 GPa are calculated (shown as Fig. 7), and the k-points of 32 32 14 and 32 32 32 are adopted for D022and L12-Ga3Zr, respectively. In the PDOS curves, the states at Fermi level are not zero, and band gap does not exist, which indicates the both phases are conductors. Furthermore, comparing with D022Ga3Zr, the states above Fermi level of L12-Ga3Zr are more significant, which suggests L12-Ga3Zr has better electrical conductivity.
For the both phases, their DOS are mainly contributed from Zr4d, Ga-4p and Ga-4s. In the plot of D022-Ga3Zr under 0 GPa (Fig. 7a), the covalent bonds between Zr and Ga atoms, especially ZreGa1 atom pairs are confirmed by the resonant peaks around 1.2 eV, 5.2 eV, 6.8 eV. These bonds mainly derive from the electron orbitals hybridization of Zr-4d and Ga(s,p) states. Similarly, in the plot of L12-Ga3Zr under 0 GPa (Fig. 7-c), the resonant peaks exist around 2.5 eV and 6.3 eV, and for the electron states below Fermi level, the electrons' localization is much insignificant than D022-Ga3Zr under 0 GPa, which suggests the covalent bonds of L12-Ga3Zr is weaker than D022-Ga3Zr. By comparing PDOS plots of D022-Ga3Zr under 0 GPa and 60 GPa (Fig. 7-a and b), one can easily find that high pressure will make the curves' energy range widened towards both negative and positive sides. Especially for the electrons states higher than Fermi level, namely for “free electrons”, high pressure will lead to lower peaks, and around Fermi level, the curves' “valley” tends to be wider and shallower. Similar case exists in the plots of L12-Ga3Zr under 0 GPa
W. Ma, J. Zhang / Journal of Alloys and Compounds 696 (2017) 1010e1018
and 60 GPa (Fig. 7-c and d). High pressure will result in a wider bandwidth, especially wider conduction band. For D022-Ga3Zr under 0 GPa (Fig. 7-a), the energy range covered under PDOS curves is about 10.5 eVe6 eV; while for D022-Ga3Zr under 60 GPa (Fig. 7-b), the energy range is widened as about 12.5 eVe8 eV. For L12-Ga3Zr under 0 GPa (Fig. 7-c), the energy range is 11.2 eVe12 eV; while for L12-Ga3Zr under 60 GPa (Fig. 7-d), the energy range is widened as about 14 eVe14 eV. Consequently, for the both phases under 60 GPa, the wider conduction bands can be confirmed, which indicates that: for both D022- and L12-Ga3Zr, there are more metallic features under 60 GPa than them under 0 GPa, and better electrical conductivity under high pressure can be anticipated. For the valence band in PDOS plots of both phases, the highest curve's peak will shift towards negative energy level. For example, from 0 GPa to 60 GPa, the peaks of D022-Ga3Zr and L12-Ga3Zr will move from 1.2 eV and 2.5 eV to 1.8 eV and 3.8 eV, respectively. This phenomenon demonstrates that the covalent bonds of Ga3Zr will form in deeper energy level under high pressure, and bring about stronger bond strength. Summarily, it is reasonable to conclude that pressure-induced delocalization of Zr electrons, especially Zr-4d, which strengthens the electron hybridization and atomic bonding between Zr and Ga atoms. Consequently, it improves the phase stability of L12-Ga3Zr, even though the phase exhibits more metallic features with the pressure increasing. 4. Summary By using DFT calculation method, the atomic structure, formation energy, elastic constants and electronic structure of D022- and L12-Ga3Zr have been investigated with respect to external pressure ranging from 0 GPa to 60 GPa. The phase stability, mechanical moduli, Debye temperature and electronic interaction are discussed. The following results are obtained. (1) Pressure induced transition from D022-Ga3Zr to L12-Ga3Zr will occur when pressure is higher than 28.48e36.34 GPa, and volume drop is about 1%. (2) The elastic constants, mechanical moduli and Debye temperature of both phases increase with pressure going up. For D022-Ga3Zr, the strengthening effect of Cij can be queued as C13 > C12 > C33 > C11 > C66 > C44; while for L12-Ga3Zr, the sequence is C44 > C12 > C11; for the mechanical moduli of both phases, the sequence is B > E > G. L12-Ga3Zr has better ductility and lower Debye temperature than D022 structure at various pressures in the range of 0e60 GPa. (3) The charge transfer from Ga to Zr exists in the both phases, and electron interaction will be more active in the circumstance with higher pressure. The Zr-4d delocalization of Ll2Ga3Zr under high pressure will strengthen the hybridization between Zr-4d and Ga(s,p), which leads to stronger atomic bonding, and subsequently makes the L12-Ga3Zr more stable under high pressure. Acknowledgements The authors acknowledge the technical support from Center for High Performance Computing of Northwestern Polytechnical University. References [1] Y. Yao, D.D. Klug, B4-B1 phase transition of GaN under isotropic and uniaxial compression, Phys. Rev. B 88 (2013) 14113.
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