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Zhi-sheng Nong, Hao-yu Wang, and Jing-chuan Zhu, First-principles calculations of structural, elastic and electronic properties of (TaNb)0.67(HfZrTi)0.33 high-entropy alloy under high pressure, Int. J. Miner. Metall. Mater., 27(2020), No. 10, pp.1405-1414. https://dx.doi.org/10.1007/s12613-020-2095-z
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To clarify the effect of pressure on a (TaNb)0.67(HfZrTi)0.33 alloy composed of a solid solution with a single body-centered-cubic crystal structure, we used first-principles calculations to theoretically investigate the structural, elastic, and electronic properties of this alloy at different pressures. The results show that the calculated equilibrium lattice parameters are consistent with the experimental results, and that the normalized structural parameters of lattice constants and volume decrease whereas the total enthalpy difference ΔE and elastic constants increase with increasing pressure. The (TaNb)0.67(HfZrTi)0.33 alloy exhibits mechanical stability at high pressures lower than 400 GPa. At high pressure, the bulk modulus B shows larger values than the shear modulus G, and the alloy exhibits an obvious anisotropic feature at pressures ranging from 30 to 70 GPa. Our analysis of the electronic structures reveals that the atomic orbitals are occupied by the electrons change due to the compression of the crystal lattices under the effect of high pressure, which results in a decrease in the total density of states and a wider electron energy level. This factor is favorable for zero resistance.
Traditional alloys, such as steel, aluminum, and magnesium alloys, are generally composed of one or two major metal elements, to which other constituent elements are added to modify the properties. In 2004, a new alloy design strategy was proposed by Yeh et al. [1] and Cantor et al. [2], which yields high-entropy alloys (HEAs). These novel alloys typically comprise five or more elements, each of which accounts for between 5at% and 35at%. Because of their unique “four core effects,” HEAs generally exhibit excellent properties such as high strength, high hardness, and good erosion and corrosion resistances [3–6]. To date, research on HEA systems divides HEAs into two main types. One is the traditional HEA, which is composed of alloying elements in the first Ⅳ cycle, such as the most widely studied alloy AlCrFeCoNi. The effect of a high strain rate on Al0.1CrFeCoNi HEA was investigated by Kumar et al. [7], who found that the formation of twin structures was related to the strain rate, with a secondary twin phase generated at high temperature. The effect of Nb content on AlCoCrFeNi HEA, as studied by Ma and Zhang [8], indicated that the structures of AlCoCrFeNi were changed with the addition of Nb, which improved the strength, hardness, and coercive force of the alloy but reduces the plasticity and saturated and residual magnetizations.
The other type of HEA is the refractory HEA composed of elements with high melting points [9–12]. This kind of HEA generally forms a single solid solution with a body-centered cubic (bcc) structure. The relationship between the microstructures and wear resistance of refractory HEAs was studied by Poulia et al. [13], who reported the presence of typical dendrite structures in MoTaWNbV alloy with a single-phase BCC structure, which was found to demonstrated excellent wear resistance. The effect of Al content on the microstructures and properties of refractory HEA was investigated by Senkov et al. [14]. The authors found that AlMo0.5NbTa0.5TiZr and Al0.4Hf0.6NbTaTiZr alloys showed extremely high compressive yield strengths above 2000 MPa at room temperature, mainly due to the effect of the solid solution strengthening between the various alloying elements. In addition to their excellent mechanical properties, refractory HEAs can also showed some unique physical properties. Guo et al. [15] reported observing zero resistance in the superconducting refractory HEA (TaNb)0.67(HfZrTi)0.33 with a single-phase BCC structure at pressures up to 190 GPa and a superconducting transition temperature (Tc) of approximately 9 K. Because of these high pressures, it is difficult to experimentally investigate the changes in the crystal structure, elastic properties, and electronic structures of the (TaNb)0.67(HfZrTi)0.33 alloy. Hence, the mechanism of superconductivity remains poorly understood. The first-principles method, in which the structural and electronic properties of a system are theoretically obtained by solving the Schrodinger equation through self-consistent calculations, can be used to calculate the total energy of the system at the electron level, so as to predict various properties. This provides an effective method for theoretically analyzing the mechanism, which is difficult to observe experimentally. The yield strength of the RhIrPdPtNiCu HEA was predicted using first-principles calculation [16], and the predicted value was found to agree well with the experimental value, which suggests the credibility of this theoretical approach.
Because the (TaNb)0.67(HfZrTi)0.33 alloy composed of Ta, Nb, Hf, Zr, and Ti elements exhibits such unique conductance properties in experiments, it provides useful guidance for the further development of functional HEAs. Hence, we chose the (TaNb)0.67(HfZrTi)0.33 alloy as the research object in this work. To theoretically explore the conductivity of the (TaNb)0.67(HfZrTi)0.33 HEA at high pressure, we used the first-principles method to calculate the structural, elastic, and electronic properties of this alloy at ground state and various pressures. In this paper, we provide an in-depth discussion of the effect of pressure on lattice parameters, elastic modulus, anisotropy, and density of states for the (TaNb)0.67(HfZrTi)0.33alloy. We also analyze the reason for the superconductivity of this alloy. The novelty of this work is its provision of the theoretical change rule for the structural, elastic, and electronic properties of (TaNb)0.67(HfZrTi)0.33 alloy at high pressure, and our explanation of the internal mechanism of superconductivity for this alloy from the view of its density of states.
The basic concept of the density functional theory (DFT) is the use of the electron density function instead of the wave function as the basic quantity for investigation, which greatly simplifies the calculation of the Schrodinger equation and provides a new path for the calculation of the physical properties of atoms, molecules, and solids at ground state. In this work, we used the first-principles method, which is based on the DFT, to calculate the total energy and the elastic and electronic properties of the (TaNb)0.67(HfZrTi)0.33 HEA. The experimental procedures we applied consisted of the following steps: (1) establishing a reasonable crystal structure model of the (TaNb)0.67(HfZrTi)0.33 HEA; (2) selecting calculated parameters to optimize the crystal structure at ground (0 K, 0 GPa) and high-pressure states, and then comparing the obtained lattice constants with the corresponding experimental results; (3) obtaining the final optimized crystal structures at the ground and high-pressure states that agree well with the experimental results by constantly adjusting the calculated parameters; (4) calculating the elastic and electronic properties of the final optimized crystal structures, and then analyzing the results using appropriate theories. In these calculations, we adopted ultrasoft pseudo-potentials [17] to describe the interaction between electrons and ions and applied the generalized gradient approximation of the Perdew–Burke–Ernzerh of parameters [18–19] to express the exchange and correlation terms. To achieve precision, we set an overall cutoff energy of 400 eV. To optimize the crystal lattice and calculate the density of states, we selected suitable grids for the k-point meshes based on the Monkhorst–Pack scheme [20–21], and determined k-point meshes of 4 × 4 × 1 after a series of convergence tests. Full relaxation of the calculations for geometric optimization and the electronic structures was achieved when the convergence thresholds were satisfied simultaneously for a maximum energy change of less than 5.0 × 10−6 eV/atom, a maximum force of less than 0.1 eV/nm, a maximum stress of lower than 0.02 GPa, and a maximum displacement of less than 5.0 × 10−5 nm. To calculate the elastic constants by the stress–strain method, we used the following convergence tolerances: a maximum energy change of <1.0 × 10−6 eV/atom, a maximum force on atoms of <0.02 eV/nm, and a maximum displacement between cycles of <1.0 × 10−5 nm. Meanwhile, for calculating the elastic constants, we set the maximum strain amplitude to 0.003. All the first-principles calculations were implemented using the Cambridge Sequential Total Energy Package (CASTEP) code.
As reported in Ref. [15], the (TaNb)0.67(HfZrTi)0.33 HEA is composed of a solid solution with a single BCC crystal structure. Therefore, we introduced a simple supercell based on a BCC unit cell to build the crystal structure of the single-phase (TaNb)0.67(HfZrTi)0.33 alloy, in which all the atoms are distributed randomly, and to ensure that the nearest-neighbor atoms are of different alloying elements [22]. We built a supercell of 72 atoms (containing 24 Ta, 24 Nb, 8 Hf, 8 Zr, and 8 Ti atoms), and the composition of this supercell ((TaNb)48(HfZrTi)24) agreed well with that of the research system ((TaNb)0.67(HfZrTi)0.33). The modeling details and assumption made regarding the atom migration and lattice deformation rules accord with those described in Ref. [23]. On this basis, the crystal structure of the (TaNb)0.67(HfZrTi)0.33 alloy in this work is a simple supercell with a tetragonal structure, with the unit cells constituting the supercell having a simple BCC structure. Random distribution of the atoms was realized by the random number method, where by a matrix, which was considered to be the set of positions for all atoms in the supercell, was randomly assigned, with different values representing different kinds of elements in the matrix. Then, a series of different supercells with random atomic occupancies were built according to the results of the random assignments. Finally, from the structures built by this random numbers method, we selected the crystal structure with the lowest energy at ground state as the initial lattice of the (TaNb)0.67(HfZrTi)0.33 alloy. To calculate the theoretical elastic constants at different pressures, we evaluated the variations in the total energy with the application of small strains on the equilibrium unit cell at a corresponding pressure. Based on the calculated theoretical elastic constants, we could estimate the polycrystalline elastic properties for the (TaNb)0.67(HfZrTi)0.33 using the Voigt–Reuss–Hill (V–R–H) approximations, as follows [24]:
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B=BV+BR2 |
(1) |
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G=GV+GR2 |
(2) |
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E=9BG3B+G |
(3) |
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ν=3B−E6B |
(4) |
where B is the bulk modulus, G is the shear modulus, E is the Young’s modulus, and ν is the Poisson’s ratio.
The Voigt and Reuss bounds for the corresponding moduli BV, GV, BR, and GR were calculated using Eqs. (5)–(8):
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BV=[(C11+C22+C33)+2(C12+C13+C23)]/9 |
(5) |
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BR=1/[(S11+S22+S33)+2(S12+S13+S23)] |
(6) |
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GV=[(C11+C22+C33−C12−C13−C23)+3(C44+C55+C66)]/15 |
(7) |
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GR=15/[4(S11+S22+S33)−4(S12+S13+S23)+3(S44+S55+S66)] |
(8) |
where Cij and Sij are the single-crystal elastic constants and the elastic compliance values of the equilibrium crystal structure at the corresponding pressure, respectively.
The elastic anisotropy, which is an important parameter that describes the possibility of inducing microcracks, can be evaluated using the elastic modulus. The percentage of elastic anisotropy for the bulk modulus AB and shear modulus AG, and the universal elastic anisotropy index AU are three common parameters for estimating the elastic anisotropy of materials. These values can be calculated using Eqs. (9)–(11) below [25]:
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AB=100×(BV−BR)/(BV+BR) |
(9) |
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AG=100×(GV−GR)/(GV+GR) |
(10) |
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AU=5GVGR+BVBR−6 |
(11) |
Generally speaking, a crystal structure with elastic isotropy is confirmed when the values of AB, AG, and AU are 0. When these values are 100% for AB and AG, this means that the structure is fully anisotropic. Regarding the universal elastic anisotropy index AU, a larger deviation from zero corresponds to a higher degree of anisotropy. In addition, the directional dependence of Young’s modulus, which indicates the variation of the Young’s modulus at different crystallographic directions, can also be used to intuitively investigate the elastic anisotropy of materials. For the crystal lattice with a tetragonal structure of the (TaNb)0.67(HfZrTi)0.33 alloy built in this work, we calculated the corresponding directional dependence of E by the following [26]:
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E=1/[(l41+l42)S11+l43S33+l21l22(2S12+S66)+l23(1−l23)(2S13+S44)] |
(12) |
where l1, l2, and l3 refer to the cosine directions with respect to the x-, y-, and z-axes, respectively.
First, we calculated the equilibrium crystal structures of the (TaNb)0.67(HfZrTi)0.33 alloy at different pressures using the Broyden–Fletcher–Goldfarb–Shanno method [27] of structure optimization, and then we obtained the corresponding equilibrium lattice constants, volume V, and total energy of the structure Et. To describe the optimized structural constants a, we introduced the arithmetic average of the lattice constants aav [23], which can be easily compared with the experimental results. The calculated optimized structural constants of the (TaNb)0.67(HfZrTi)0.33 alloy at zero pressure is a0 = 0.3398 nm, and the previously obtained experimental values are aexp ≈ 0.333–0.343 nm [28]. The normalized structural parameters of a/a0 and V/V0 at a pressure of 100 GPa, as obtained by first-principles calculations, are also consistent with the experimental results [15]. This suggests the high precision and reliability of these calculations. The pressure dependence of a/a0, V/V0, and the total enthalpy difference ΔEt = Et– Et,0 are plotted in Fig. 1, in which a0, V0, and Et,0 are arithmetic averages of the lattice constants, volume, and total enthalpy of the equilibrium structure at 0 GPa, respectively. It is clear that the equilibrium ratios of a/a0 and V/V0 decrease monotonically with increases in pressure, whereas the total enthalpy difference ΔEt shows an increasing trend. This indicates that the crystal structure of the (TaNb)0.67(HfZrTi)0.33 alloy is continuously compressed with increases in pressure, which results in the decreasing stability of the crystal structure. Further experimental data on the total energies of the crystals are needed to verify this conclusion.
The elastic constants Cij, which link the mechanics and dynamics, can provide important information about the reliability of structural models and engineering applications. In the present work, after full relaxation, we found each unit cell have an orthorhombic structure in the supercell, whereas the optimized crystal structure of the supercell for the (TaNb)0.67(HfZrTi)0.33 alloy had a generalized tetragonal structure. Hence, we obtained six independent elastic constants for the crystal structure of the (TaNb)0.67(HfZrTi)0.33 alloy using the stress–strain relation, and Fig. 2 shows these elastic constants as a function of pressure. We note that except for C44 and C66, the elastic constants all increase linearly with pressure. C11 and C33 increase more rapidly with pressure, which implies that the change of C11 and C33 are the most sensitive to pressure, and C44 and C66 are found to be more unresponsive than the others.
For a mechanically stable crystal structure, the corresponding Born stability criteria for a tetragonal crystal at P GPa are as follows [29]:
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{C11>0C33>0C44>0C66>0 |
(13) |
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C11−C12>0 |
(14) |
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C11+C33−2C13>0 |
(15) |
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2C11+C33+2C12+4C13>0 |
(16) |
In Fig. 2, we can easily see that Cij satisfy the Born stability criteria of Eqs. (13) and (16) because of the positive values of Cij at P pressure. To further investigate the mechanical stability of the (TaNb)0.67(HfZrTi)0.33 alloy at high pressure, we plotted the mechanical stability criteria of Eqs. (14) and (15) for (TaNb)0.67(HfZrTi)0.33 as a function of pressure, as shown in Fig. 3. Although the calculated values of Eqs. (14) and (15) do not increase monotonically, these values are all higher than 0 (the value of Eq. (14) at a pressure of 50 GPa is about 27.7 GPa), which indicates that Cij also satisfy the Born stability criteria of Eqs. (14) and (15). Therefore, the crystal structures of the (TaNb)0.67(HfZrTi)0.33 alloy in this work are mechanically stable at high pressures less than 400 GPa.
The polycrystalline elastic modulus, including the bulk modulus, shear modulus, and Young's modulus, play important roles in the technological characterization of materials. In this work, we calculated the elastic modulus using the V–R–H approximations. Fig. 4 shows the corresponding pressure dependences of B, G, and E for (TaNb)0.67(HfZrTi)0.33HEA, in which we can see that the bulk modulus obviously increases linearly with increasing pressure. This implies that the resistance to volume change of the crystal structure of (TaNb)0.67(HfZrTi)0.33 alloy is continually enhanced at high pressure. Regarding the shear modulus and Young's modulus, the calculated values of G and E increase steadily at pressures higher than 70 GPa. However, when the pressure is lower than 70 GPa, the G and E curves first rise and then fall, but the minimum values for G and E between 0 to 50 GPa are still 11.3 and 33.5 GPa, respectively. We speculate that the resistance to reversible deformations under shear stress and the resistance against uniaxial tensions would fluctuate at low pressure (<70 GPa).The values of B are all larger than those of G, which indicates that the shear modulus is the stability-limiting parameter of the (TaNb)0.67(HfZrTi)0.33 alloy [30].
The brittle–ductile behaviors of materials can be evaluated based on their calculated polycrystalline elastic properties. The intrinsic ductility and brittleness of materials can be predicted using an important comprehensive criterion that includes the ratio of bulk modulus to shear modulus B/G, Poisson’s ratio ν, and the Cauchy pressure C12–C44 [31–33]. Usually, a ductile material can be obtained when B/G > 1.75, ν > 0.26, and C12–C44 > 0 are concurrently satisfied. Otherwise, a material exhibits obvious brittleness. Fig. 5 shows the pressure dependence of B/G, ν, and C12–C44 for the (TaNb)0.67(HfZrTi)0.33 alloy. As we can see, the minimum values of B/G and ν for the (TaNb)0.67(HfZrTi)0.33 alloy at pressure of 10 GPa are 4.78 and 0.40, respectively, whereas the minimum value of 83.6 GPa for C12–C44 occurs at ground state (0 K, 0 GPa). This suggests that B/G, ν, and C12–C44 for the (TaNb)0.67(HfZrTi)0.33 alloy all meet the criteria for ductility, which means that this alloy will be have in a ductile manner at high pressure. We also note that the values of B/G and ν reach a sudden maximum at about 50 GPa, which implies that the ductility of (TaNb)0.67(HfZrTi)0.33 would markedly improve at a pressure of about 50 GPa. This is associated with a decrease in the shear modulus at this pressure (shown in Fig. 4).
Next, we evaluated the elastic anisotropy of the (TaNb)0.67(HfZrTi)0.33 alloy, including the percentages of the bulk modulus and shear modulus, and the universal elastic anisotropy index. The corresponding directional dependences of Young's modulus at different pressures were also calculated. Fig. 6 shows these elastic anisotropy parameters for the (TaNb)0.67(HfZrTi)0.33 alloy as a function of pressure, from which we find that the AG values tend to be stable and are clustered around 0.5 to 1.5 when the pressures are greater than 70 GPa, with the maximum value of 75.0 occurring at a pressure of 40 GPa. Regarding AB, although there is no obvious variation rule with increases in pressure, two maxima occur at 1.95 and 0.67 at pressures of 0 and 370 GPa, respectively. In addition, the values of AB are lower than those of AG at pressures from 0 to 400 GPa, which indicates that the elastic anisotropy of the shear modulus AG plays the dominant role in the anisotropic properties of the alloy in this pressure range. The universal elastic anisotropy index AU, which combines the influences of AB and AG, can be used to evaluate the overall elastic anisotropy of the materials. In Fig. 6, we can see that the pressure dependence of AU has a similar change rule to that of AG, and values higher than 1.0 for AU are observed at pressures from 30 to 70 GPa, as well as at 0 GPa. This implies that the mechanical behavior of the (TaNb)0.67(HfZrTi)0.33 alloy is anisotropic at pressures between 30 and 70 GPa, as well as at 0 GPa. When the pressure is lower than 30 GPa or higher than 70 GPa, the AU values decrease to about 0.20, which indicates obvious weakening of the elastic anisotropy.
To visually investigate the in-plane elastic anisotropy, we introduced the directional dependence of E, as shown in Fig. 6. For comparison with the AU results, we studied the directional dependence of the E of the crystal structures at pressures of 0, 40, 100, 190, 280, 310, 370, and 400 GPa. Generally speaking, the spherical shape of the three-dimensional directional dependence of E corresponds to a fully isotropic structure. Clearly, the surface representations of the (TaNb)0.67(HfZrTi)0.33 alloy at pressures of 190, 280, 310, 370, and 400 GPa are closer to being spherical than the others, which implies that the alloy is more isotropic when the pressure is greater than 100 GPa. This is consistent with the AB, AG, and AU results discussed above.
To investigate the effect of pressure on the electronic structures of the (TaNb)0.67(HfZrTi)0.33 alloy, we calculated the total density of states (TDOS) and plotted the results in Fig. 7. We can see that the TDOS energy regions can be separated into three categories, i.e., lower-energy regions (from −58 to −46 eV), low-energy regions (from −36 to −24 eV), and Fermi-level regions (from −10 to 6 eV). A notable difference in the TDOS for the (TaNb)0.67(HfZrTi)0.33 alloy at different pressures occurs with increasing pressure to 100 GPa, for which the peaks become lower- and the low-energy regions shift toward the higher-energy direction, whereas the peaks located in the Fermi-level regions shift toward the lower energy direction. When the pressure is higher than 100 GPa, the TDOS values in the low-energy regions obviously decrease, and the peaks at roughly from −36 and −28 eV overlap. The reason for this is likely to be that the distance between the atoms decreases under the effect of high pressure, which leads to a change in the atomic orbitals occupied by electrons, which results in the appearance of new hybridizations between different elements. We discuss the specific hybridizations of elements below in our discussion of the partial density of states.
In addition, at the Fermi level, the TDOS for the (TaNb)0.67(HfZrTi)0.33 alloy decreases with increasing pressure, accompanied by a widening of the electron energy level. This can be attributed to two aspects, one of which is the appearance of additional electrons with mutual repulsion because of the reduction in atomic distance, which leads to an increase in the electron energy levels. The second aspect is that as the distance between atoms decreases, electrons are easily attracted to a new steady-state energy level, and the corresponding energy level of the electrons is lower with decreasing distance. The pressure dependence of the TDOS reduction rate at the Fermi level R(EF) is relative to the pressure of 0 GPa, as shown in Fig. 8. R(EF) = {[N(EF, 0) −N(EF, P)]/N(EF, 0)}, in which N(EF, P) refers to the total density of states at the Fermi level (EF) under the pressure of P GPa.
We found N(EF, P) in the (TaNb)0.67(HfZrTi)0.33 alloy to decrease with increases in pressure. Based on the Bardeen–Cooper–Schrieffer superconducting theory [34], a decrease in the TDOS at the Fermi level indicates that Tc would be lower with increases in pressure. We speculate that Tc for the (TaNb)0.67(HfZrTi)0.33 alloy would be lower with increasing pressure, which agrees well with experimental results obtained at pressures over 60 GPa [15]. We also note that there are two transition points in the reduction velocity, corresponding to 150 and 310 GPa, respectively. It can be assumed that increases in pressure cause the lattice to be continually compressed, and electrons would be held together by Coulomb forces, perhaps when the lattice changes to 0.62V0. These clusters of electrons would play a role similar to that of superconducting electrons in the conduction process, in other words, the momentum change of these clusters of electrons is always 0, which is a major factor in the zero resistance of alloys. Therefore, the appearance of these two transition points in the reduction velocity likely indicates that free electrons are converted into clusters of electrons by tight atoms bound under the effect of high pressures. This is likely to again increase the Tc for the (TaNb)0.67(HfZrTi)0.33 alloy when the pressure is higher than 310 GPa. Although alloys will inevitably break at such a high pressure, these results can still be used to guide the design of zero-resistance alloys. In addition, the appearance of transition points in the reduction velocity for N(EF, P) implies that there may be Lifshitz transitions (change in the Fermi surface topology) in the vicinity of these two pressures for the (TaNb)0.67(HfZrTi)0.33 alloy [35]. Of course, more detailed information and eventual confirmation will require further research.
We also calculated the partial density of states (PDOS), which can provide important information on the hybridization of different elements and the bonding behavior of alloys. Fig. 9 shows the corresponding PDOS values for the (TaNb)0.67(HfZrTi)0.33 alloy at pressures of 0, 150, and 310 GPa, in which we can easily see that the TDOS values for (TaNb)0.67(HfZrTi)0.33 at 0 GPa are mainly dominated by the s states of elements Nb, Zr, and Ti in the lower-energy regions from −58 to −46 eV, and the Ta-p and Hf-p states start contributing to the TDOS in these energy regions with increases in pressure. In the low-energy regions (from −36 to −24 eV), the p states of elements Nb, Zr, and Ti provide the main contributions, but the Ta-p and Hf-p states also play a role at pressures of 150 and 310 GPa. In addition, the peaks of Nb-p, Zr-p, and Ti-p are obviously wider and even overlap at high pressure, as discussed above, which can also be attributed to the reduced atomic distance. When the pressure is higher than 150 GPa, obvious hybridizations occur between the s states of elements Nb and Ti and the p states of elements Ta and Hf from −58 to −46 eV, and additional p-p hybridizations are also found in the low-energy regions (from −36 to −24 eV). This implies the formation of covalent bonding in the (TaNb)0.67(HfZrTi)0.33 alloy at high pressure. We note that some new stable bondings occur between elements Nb, Zr, and Ti (Fig. 9(b)) in the energy regions ranging from−36 to −24 eV at a pressure of 150 GPa. At the same time, the electron energy levels associated with these new bondings are all lower than those of the initial bondings among elements Nb, Zr, and Ti. These new element bondings cause the electron cloud to form an obvious forbidden band, which implies easier electron transition and filling. Hence, the pattern of electron distribution has changed in which electrons are distributed into any energy region at high pressure rather than filling from low energy to high energy for the (TaNb)0.67(HfZrTi)0.33 alloy. This is also a favorable factor for zero resistance.
Although the structural, elastic, and electronic properties of the (TaNb)0.67(HfZrTi)0.33 alloy at different pressures have been obtained, along with some interesting results, the crystal structure of the (TaNb)0.67(HfZrTi)0.33 alloy, which was built using a simple supercell based on bcc unit cells, remains a long-range goal and serves as an approximation method for simulating real HEAs. To conduct an in-depth computational investigation of HEAs, other approaches such as the KKR–CPA (Korringa–Kohn–Rostoker (KKR)–coherent potential approximation) technique [35] must be further developed and applied.
In conclusion, in this work, we investigated the structural, elastic, and electronic properties of the (TaNb)0.67(HfZrTi)0.33 high-entropy alloy at different pressures using first-principles calculations. The equilibrium ratios a/a0 and V/V0 were found to decrease monotonically with increases in pressure, whereas the total enthalpy difference E−E0 and elastic constants showed an increasing trend. The bulk modulus was found to increase linearly with increasing pressure, and to show larger values than the shear modulus. The ductility of the (TaNb)0.67(HfZrTi)0.33 alloy improved markedly when the pressure was approximately 50 GPa. The calculated anisotropy values at pressures of 0 and 30–70 GPa showed obvious anisotropic features, whereas the elastic anisotropy of the alloy was weakened at other pressures. The peaks of the TDOS for the (TaNb)0.67(HfZrTi)0.33 alloy were observed to shift in the higher energy direction, whereas the peaks in the Fermi-level regions shifted in the descending energy direction. The distance between atoms was determined to decrease under the effect of high pressure, there by leading to changes in the atomic orbitals occupied by electrons. This results in the appearance of new hybridizations between different elements in the energy regions ranging from −36 to−24 eV, and is favorable for zero resistance. Finally, according to our analysis of the change in properties and our explanation of the internal mechanism of superconductivity for the (TaNb)0.67(HfZrTi)0.33 alloy at high pressure, we predict that the conductance properties of functional high-entropy alloys can be regulated by controlling the TDOS value at the Fermi level.
This work was financially supported by the National Natural Science Foundation of China (No. 51701128) and the Scientific Research Project of Education Department of Liaoning Province, China (No. JYT19037).
|
J.W. Yeh, S.K. Chen, S.J. Lin, J.Y. Gan, T.S. Chin, T.T. Shun, C.H. Tsau, and S.Y. Chang, Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes, Adv. Eng. Mater., 6(2004), p. 299. DOI: 10.1002/adem.200300567
|
|
B. Cantor, I.T.H. Chang, P. Knight, and A.J.B. Vincent, Microstructural development in equiatomic multicomponent alloys, Mater. Sci. Eng. A, 375-377(2004), p. 213. DOI: 10.1016/j.msea.2003.10.257
|
|
M. Vaidya, K. Guruvidyathri, and B.S. Murty, Phase formation and thermal stability of CoCrFeNi and CoCrFeMnNi equiatomic high entropy alloys, J. Alloys Compd., 774(2019), p. 856. DOI: 10.1016/j.jallcom.2018.09.342
|
|
D.B. Miracle and O.N. Senkov, A critical review of high entropy alloys and related concepts, Acta Mater., 122(2017), p. 448. DOI: 10.1016/j.actamat.2016.08.081
|
|
Y. Zhang, T.T. Zuo, Z. Tang, M.C. Gao, K.A. Dahmen, P.K. Liaw, and Z.P. Lu, Microstructures and properties of high-entropy alloys, Prog. Mater. Sci., 61(2014), p. 1. DOI: 10.1016/j.pmatsci.2013.10.001
|
|
K.A. Christofidou, E.J. Pickering, P. Orsatti, P.M. Mignanelli, T.J.A. Slater, H.J. Stone, and N.G. Jones, On the influence of Mn on the phase stability of the CrMnxFeCoNi high entropy alloys, Intermetallics, 92(2018), p. 84. DOI: 10.1016/j.intermet.2017.09.011
|
|
N. Kumar, Q. Ying, X. Nie, R.S. Mishra, Z. Tang, P.K. Liaw, R.E. Brennan, K.J. Doherty, and K.C. Cho, High strain-rate compressive deformation behavior of the Al0.1CrFeCoNi high entropy alloy, Mater. Des., 86(2015), p. 598. DOI: 10.1016/j.matdes.2015.07.161
|
|
S.G. Ma and Y. Zhang, Effect of Nb addition on the microstructure and properties of AlCoCrFeNi high-entropy alloy, Mater. Sci. Eng. A, 532(2012), p. 480. DOI: 10.1016/j.msea.2011.10.110
|
|
O.N. Senkov, J.M. Scott, S.V. Senkova, D.B. Miracle, and C.F. Woodward, Microstructure and room temperature properties of a high-entropy TaNbHfZrTi alloy, J. Alloys Compd., 509(2011), No. 20, p. 6043. DOI: 10.1016/j.jallcom.2011.02.171
|
|
C.M. Liu, H.M. Wang, S.Q. Zhang, H.B. Tang, and A.L. Zhang, Microstructure and oxidation behavior of new refractory high entropy alloys, J. Alloys Compd., 583(2014), p. 162. DOI: 10.1016/j.jallcom.2013.08.102
|
|
Y. Zou, S. Maiti, W. Steurer, and R. Spolenak, Size-dependent plasticity in an Nb25Mo25Ta25W25 refractory high-entropy alloy, Acta Mater., 65(2014), p. 85. DOI: 10.1016/j.actamat.2013.11.049
|
|
H.W. Yao, J.W. Qiao, J.A. Hawk, H.F. Zhou, M.W. Chen, and M.C. Gao, Mechanical properties of refractory high-entropy alloys: Experiments and modeling, J. Alloys Compd., 696(2017), p. 1139. DOI: 10.1016/j.jallcom.2016.11.188
|
|
A. Poulia, E. Georgatis, A. Lekatou, and A.E. Karantzalis, Microstructure and wear behavior of a refractory high entropy alloy, Int. J. Refract Met. Hard Mater., 57(2016), p. 50. DOI: 10.1016/j.ijrmhm.2016.02.006
|
|
O.N. Senkov, S.V. Senkova, and C. Woodward, Effect of aluminum on the microstructure and properties of two refractory high-entropy alloys, Acta Mater., 68(2014), p. 214. DOI: 10.1016/j.actamat.2014.01.029
|
|
J. Guo, H.H. Wang, F.Von Rohr, Z. Wang, S. Cai, Y.Z. Zhou, K. Yang, A.G. Li, S. Jiang, Q. Wu, R.J.Cava, and L.L. Sun, Robust zero resistance in a superconducting high-entropy alloy at pressures up to 190 GPa, Proc. Nat. Acad. Sci. U.S.A., 114(2017), No. 50, p. 13144. DOI: 10.1073/pnas.1716981114
|
|
B.L. Yin and W.A. Curtin, First-principles-based prediction of yield strength in the RhIrPdPtNiCu high-entropy alloy, npj Comput. Mater., 5(2019), art. No. 14. DOI: 10.1038/s41524-019-0151-x
|
|
D. Vanderbilt, Soft self-consistent pseudopotentials in a generalized eigen value formalism, Phys. Rev. B, 41(1990), No. 11, p. 7892. DOI: 10.1103/PhysRevB.41.7892
|
|
M. Marlo and V. Milman, Density-functional study of bulk and surface propertiesof titanium nitride using different exchange-correlation functionals, Phys. Rev. B, 62(2000), p. 2899. DOI: 10.1103/PhysRevB.62.2899
|
|
J.A. White and D.M. Bird, Implementation of gradient-corrected exchange-correlation potentials in Car-Parrinello total-energy calculations, Phys. Rev. B, 50(1994), No. 7, p. 4954. DOI: 10.1103/PhysRevB.50.4954
|
|
J.D. Pack and H.J. Monkhorst, Special points for Brillouin-zone integrations-areply, Phys. Rev. B, 16(1977), No. 4, p. 1748. DOI: 10.1103/PhysRevB.16.1748
|
|
H.J. Monkhorst and J.D. Pack, Special points for Brillouin-zone integrations, Phys. Rev. B, 13(1976), No. 12, p. 5188. DOI: 10.1103/PhysRevB.13.5188
|
|
E. Delczeg-Czirjak, E. Nurmi, K. Kokko, and L. Vitos, Effect of long-range order on elastic properties of Pd0.5Ag0.5 alloy from first principles, Phys. Rev. B, 84(2011), No. 9, art. No. 094205. DOI: 10.1103/PhysRevB.84.094205
|
|
Z.S. Nong, J.C. Zhu, and R.D. Zhao, Prediction of structure and elastic properties of AlCrFeNiTi system high entropy alloys, Intermetallics, 86(2017), p. 134. DOI: 10.1016/j.intermet.2017.03.014
|
|
H.Z. Yao, L.Z. Ouyang, and W.Y. Ching, Ab initio calculation of elastic constants of ceramic crystals, J. Am. Ceram. Soc., 90(2007), No. 10, p. 3194. DOI: 10.1111/j.1551-2916.2007.01931.x
|
|
L.S. Ma, Y.H. Duan, and R.Y. Li, Structural, elastic and electronic properties of C14-type Al2M (M = Mg, Ca, Sr and Ba) Laves phases, Physica B, 507(2017), p. 147. DOI: 10.1016/j.physb.2016.12.004
|
|
J.F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford, 1985.
|
|
C.G. Broyden, J.E. Dennis, and J.J. Moré, On the local and superlinear convergence of Quasi-Newton methods, JMA Appl. Math., 12(1973), No. 3, p. 223.
|
|
F. von Rohr, M.J. Winiarski, J. Tao, T. Klimczuk, and R.J. Cava, Effect of electron count and chemical complexity in the Ta–Nb–Hf–Zr–Ti high-entropy alloy superconductor, Proc. Nat. Acad. Sci. U.S.A., 113(2016), No. 46, p. 7144. DOI: 10.1073/pnas.1615926113
|
|
M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954.
|
|
I.R. Shein and A.L. Ivanovskii, Elastic properties of mono-and polycrystalline hexagonal AlB2-like diborides of s, p and d metals from first-principles calculations, J. Phys.:Condens. Mater., 20(2008), No. 41, art. No. 415218. DOI: 10.1088/0953-8984/20/41/415218
|
|
S.F. Pugh, XCII. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals, London Edinburgh Dublin Philos. Mag. J. Sci., 45(1954), No. 367, p. 823. DOI: 10.1080/14786440808520496
|
|
P. Li, L.S. Ma, M.J. Peng, B.P. Shu, and Y.H. Duan, Elastic anisotropies and thermal conductivities of WB2 diborides in different crystal structures: A first-principles calculation, J. Alloys Compd, 747(2018), p. 905. DOI: 10.1016/j.jallcom.2018.03.109
|
|
H.S. Li, Y. Cao, S.G. Zhou, P.X. Zhu, and J.C. Zhu, Site preferences and effects of X (X = Mn, Fe, Co, Cu) on the properties of NiAl: A first-principles study, Mod. Phys. Lett. B, 30(2016), No. 9, art. No. 1650133. DOI: 10.1142/S0217984916501335
|
|
J. Bardeen, L.N. Cooper and J.R. Schrieffer, Theory of superconductivity, Phys. Rev., 108(1957), No. 5, p. 1175. DOI: 10.1103/PhysRev.108.1175
|
|
K. Jasiewicz, B. Wiendlocha, K. Górnicka, K. Gofryk, M. Gazda, T. Klimczuk, and J. Tobola, Pressure effects on the electronic structure and superconductivity of (TaNb)0.67(HfZrTi)0.33 high entropy alloy, Phys. Rev. B, 100(2019), No. 18, art. No. 184503. DOI: 10.1103/PhysRevB.100.184503
|
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