Cite this Article
Liu Jun-Chao, Yuan Zhi-Hong, Li Shi-Chang, Kong Xiang-Gang, Yu You, Ma Sheng-Gui, Sang Ge, Gao Tao. Structural, electronic, vibrational, and thermodynamic properties of : A first-principles-based study. Chinese Physics B, 2018, 27(4): 047802  
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Structural, electronic, vibrational, and thermodynamic properties of : A first-principles-based study
Liu Jun-Chao1, 2, Yuan Zhi-Hong1, Li Shi-Chang1, Kong Xiang-Gang1, Yu You3, Ma Sheng-Gui1, Sang Ge4, Gao Tao1, †
Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
College of Physical Science and Technology, Sichuan University, Chengdu 610065, China
College of Optoelectronic Technology, Chengdu University of Information Technology, Chengdu 610225, China
Science and Technology on Surface Physics and Chemistry Laboratory, P. O. Box 9071-35, Jiangyou 621907, China

 

† Corresponding author. E-mail: gaotao@scu.edu.cn

Abstract

The physical properties including structural, electronic, vibrational and thermodynamic properties of Zr1−xHfxCo (x is the concentration of constituent element Hf, and changes from 0 to 1) are investigated in terms of the ABINIT program. The results reveal that all of Zr1−xHfxCo have similar physical properties. When Hf concentration x gradually increases from 0.0 to 1.0, the lattice constant decreases from 3.217 Å to 3.195 Å very slowly. The calculated density of states (DOS) indicates that the metallic nature is enhanced and the electrical conductivity turns better with the increase of Hf. Moreover, as Hf concentration increases from 0 to 1, the Fermi energy gradually increases from −6.96 eV to −6.21 eV, and the electronic density of states at the Fermi level decreases from 2.795 electrons/eV f.u. down to 2.594 electrons/eV f.u., both of which imply the decrease of chemical stability. The calculated vibrational properties show that the increase of Hf concentration from 0 to 1 causes the maximum vibrational frequency to decrease gradually from about 223 cm−1 to 186 cm−1, which suggests a lower dispersion gradient and lower phonon group velocities for these modes. Finally, the phonon related thermodynamic properties are obtained and discussed.

PACS: ;78.30.Er;;21.65.-f;;21.60.De;
Keyword:;Zr1-xHfxCo;;electronic properties;;vibrational properties;;thermodynamic properties;
1. Introduction

The intermetallic compound ZrCo has received considerable attention because it is a probable candidate material for the storage and delivery of hydrogen isotopes in the international thermonuclear experimental reactor team (ITER). Compared with uranium conventionally serving as a getter in ITER, ZrCo has some inherent advantages, e.g., no radioactivity, less pyrophoricity, easier to operate. Moreover, ZrCo has excellent hydriding/dehydriding properties, such as the extremely low equilibrium absorption pressure (down to 10−2 Pa at room temperature for tritium recovery), and the not too high dehydrogenation temperature (approximately 400 °C when dissociation hydrogen pressure reaches 1 atm, 1 atm = 1.01325×105 Pa).[1–7] Therefore, ZrCo has been selected and used to store and transport hydrogen isotopes by ITER.[8,9]

Many experimental researches on the Zr1−xMxCo system in which Zr atom is partially substituted for by other kind of atom M, such as Ti and Hf, confirm that some kind of element substitution can inhibit the disproportionation reaction, which is easily induced in the course of the hydrogen recovery/delivery processes of ZrCo thereby making ZrCo lose its ability to store hydrogen isotopes.[5,10–15]

However, to the best of our knowledge, no theoretical ab-initio study on the Zr1−xMxCo system has been reported except the ZrCo and HfCo. Li et al.[16] investigated the vibrational and thermodynamic properties of ZrCo by density functional perturbation theory (DFPT). Chattaraj et al.[17,18] reported the structural, electronic, vibrational, thermodynamic and elastic properties of ZrCo by using a first principles approach within the Vienna ab initio simulation package (VASP). In addition, Chattaraj et al.[19] measured the enthalpy increments of Zr0.5Co0.5 in a temperature range of 642 K–1497 K, then the thermodynamic functions of Zr0.5Co0.5 in a temperature range of 298.15 K–1500 K were obtained through the measuring enthalpy increments. Recently, Lu et al.[20] investigated structural, elastic, and electronic properties of CoZr under high pressure by first-principles study, and confirmed that CoZr is mechanically stable at various pressures and the structural stability decreases under high pressure.

Here, we aim at studying the structural, electronic, dynamic, and thermodynamic properties of Zr1−xHfxCo (x is the concentration of constituent element Hf, and changes from 0 to 1 in steps of , the x has the same meaning in this paper, unless otherwise stated) by first-principles study.

The rest of this paper is arranged as follows. In Section 2, we give a detailed description of the computational method. In Section 3, the acquired results and their interpretation are given. Finally, a brief summary is given in Section 4.

2. Computational details

All the calculations in this work are performed through the density functional theory (DFT) and DFPT in the plane-wave pseudopotential approach by using the ABINIT code.[21] The generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE)[22] is adopted for exchange–correlation energy, and the electronic pseudo-potentials are used for atoms Zr-4s24p64d25s2, Co-3s23p63d74s2, and Hf-5s25p65d26s2, respectively. According to our convergence test, an 8×8×8 Monkhorst-pack k-point mesh in the Brillouin zone for one unit cell is used, and the energy cut-off is set with 30 Hartree (1 Hartree = 4.3597×10−18 J) to determine the number of plane waves in expansion which guarantees the total energy errors within 10−10 Hartree in all our calculations. After structural optimization, the detailed vibrational properties including the phonon dispersion curves, phonon density of states, and phonon frequencies at zone-center are obtained by the DFPT. Thereafter, thermodynamic properties such as internal energy E, entropy S, and specific heat CV are predicted within the harmonic approximation, then Gibbs free energy G is computed through the obtained internal energy E and entropy S.

The virtual crystal approximation (VCA)[23] implemented in the ABINIT program is used to simulate the disordered alloy Zr1 −xHfxCo. Two main approaches exist for treating the disordered alloy within first principles methods: the supercell (SC) method and the VCA method. The SC method can give more correct results but the calculation time becomes rapidly too large compared with the VCA method.[24] In contrast to this, the VCA method has the advantages of simplicity and computational efficiency because the VCA allows the crystal to remain the primitive periodicity but contains some fictitious “virtual” atoms whose potential is the average of potentials of the atoms in the parent compounds.[24–26] So, the VCA method is an alternative and a promising approach to treating the disordered alloys and has successfully predicted the physical properties, including structural, electronic, piezoelectric, elastic, magnetic, metal, and dynamic properties of many disordered alloy systems.[24,25,27–32]

Therefore, we believe that the VCA method is a reliable method of predicting the electronic, vibrational, and thermodynamic properties of the ternary Zr1−xHfxCo alloy. In the Zr1 −xHfxCo system, the virtual pseudo-potential of a virtual atom is obtained by the linear combination of elemental ionic pseudo-potentials of ZrCo and HfCo within first-principles VCA scheme, that is to say, , where x is the Hf substitute fraction.

3. Results and discussion
3.1. Structural properties

Experiments[15,33,34] have confirmed that solid solution Zr1−xHfxCo (x=0.1, 0.2, 0.25, 0.3, 0.5, 0.75) are rather stable and that crystal structures of Zr1−xHfxCo are identical with those of alloys ZrCo and HfCo. That is to say, Zr1−xHfxCo (x=0.1, 0.2, 0.25, 0.3, 0.5, 0.75) are of CsCl type simple cubic structure with space group of (No. 221).[35] Therefore, we believe that all of Zr1−xHfxCo ( ) have the same crystal structures as ZrCo and HfCo. In the subsequent calculation, we work starting from the above structure properties.

The equilibrium lattice parameter is computed from the structural optimization, using the Broyden–Fletcher–Goldfarb–Shanno minimization.[37–40] The results of optimization for Zr1−xHfxCo along with other theoretical and experimental results are listed in Table 1. It can be seen that our calculated lattice constants for Zr1−xHfxCo are very close to other theoretical[16,17] and experimental results,[34,36] which ensures the accuracy and the reliability of our further study of Zr1−xHfxCo.

Table 1.

Optimized and experimental lattice constants of Zr1−xHfxCo, x represents the concentration of constituent element Hf.

.

From Table 1, we can see that the lattice constant gradually decreases from 3.227 Å to 3.195 Å with the increase of the Hf substitute rate, but change is quite small. A slight change of lattice constant is logical for the reason that atom Hf and atom Zr have similar valence electronic configurations and almost the same atom radius. The decrease of lattice constant means that the crystal plane spacing of ZrCo is bigger than that of HfCo, which is in agreement with experimental results reported by Peng et al.[15]

3.2. Electronic properties

In order to understand the effect of Hf-doping on the electronic behavior of Zr1−xHfxCo, we calculate the density of states (DOS) of Zr1−xHfxCo around the Fermi level, some of them are shown in Fig. 1. The calculated results reveal that Zr1−xHfxCo have similar DOSs for different Hf substitute ratios, which implies that all of Zr1−xHfxCo exhibit similar electronic properties. The reason for this phenomenon is that element Hf and element Zr belong to the same family, atom Hf and atom Zr have similar valence electronic configurations. Further analysis shows that above the Fermi level, with the increase of Hf concentration from 0 to 1, the energy range expands and the peak value (around 4 eV) of DOS decreases from 6.085 electrons/eV f.u. to 4.235 electrons/eV f.u. That is to say, the DOS shifts towards the high energy and the Fermi surface moves down with Hf content increasing. All these phenomena show that the nature of metal is strengthened with increasing Hf content. Consequently, it can be predicted that as the Hf concentration gradually increases, the conductivity is getting better.[20,32]

Fig. 1. (color online) Plots of DOS versus energy for Zr1−xHfxCo with x=0.0, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0.

Fermi energy and the electron density of states at the Fermi level can describe the stability of the compound and the strength of electron interaction. The Fermi energy and the electronic density of states at the Fermi level are given in Fig. 2. Figure 2(a) shows that the higher the concentration of Hf, the higher the Fermi energy is, which induces the lower chemical stability of the compound and lattice contraction. Moreover, one can see from Fig. 2(b) that the decreases with the increase of Hf content, which implies that orbital hybridization between Zr1−xHfx and Co in the region near the Fermi level and the electronic interaction of the atomic orbital weaken. This phenomenon also indicates that the chemical stability decreases as the Hf content increases.[17,20]

Fig. 2. (color online) (a) Hf concentration dependence of Fermi energy. (b) Hf concentration dependence of TDOS at Fermi level .
3.3. Vibrational properties

Herein, the detailed vibrational properties of Zr1 −xHfxCo are obtained by the DFPT. The calculated phonon dispersion curves (along the principal symmetry direction of the Brillouin zone ) and the corresponding total phonon density of states for Zr1 −xHfxCo (x=0.0, 0.5, 1.0) are plotted in Fig. 3. Evidently, an increase in the Hf content leads to the phonon frequency decreasing. Since the primitive cell of Zr1−xHfxCo contains two atoms, the corresponding numbers of vibrational modes are six, of which three are acoustic modes and the remaining modes are optical ones. According to the standard group-theoretical analysis, we can obtain the following irreducible representation at the Γ point:

where the IR represents infrared active modes, subscript u denotes the antisymmetric mode with respect to the center of inversion, and is the three-fold degenerate mode. The calculated phonon frequencies at the Γ point are listed in Table 2. As the Hf concentration gradually increases from 0.0 to 1.0, the calculated phonon frequencies of the optical modes at the Γ point gradually decrease from 184.28 cm−1 to 173.18 cm−1, which is due to the mass effect. Unfortunately, we are not aware of any experimental reports on the vibrational properties of Zr1−xHfxCo, but our results of ZrCo are in agreement with the other theoretical results.[18]

Fig. 3. (color online) Calculated phonon dispersion curves and the phonon densities of states for (a) ZrCo, (b) Zr0.5Hf0.5Co, and (c) HfCo.
Table 2.

Calculated optical mode at the Γ point of Zr1−xHfxCo, (x represents the concentration of constituent element Hf).

.

Figure 4 gives the calculated phonon density of states (PHDOS) for Zr1−xHfxCo (x=0.0, 0.2, 0.4, 0.5, 0.6, 0.8, and 1.0). Two main peaks appear in Fig. 4, peak I significantly moves towards a low wave number with the increase of Hf concentration. Additionally, the increase of Hf concentration from 0 to 1 leads to the highest frequency mode gradually decreasing from about 223 cm−1 down to about 186 cm−1, which implies a lower dispersion gradient and then lower phonon group velocities for these modes.[41] The reason for this phenomenon is that the atomic weight of the Hf atom is 1.96 times larger than that of the Zr atom, which leads to the increase of the reduced mass, therefore the vibrational frequency will decrease with the content of Hf increasing.

Fig. 4. (color online) Plots of PHDOS versus frequency of Zr1−xHfxCo.

On the contrary, peak II of the phonon density of states slightly shifts towards a high wave number, which is due to the fact that vibrational frequency relates to the reduced mass of Zr1 −xHfxCo and interaction between Zr1−xHfx and Co. Specifically, the stronger interaction between Zr1−xHfx and Co will result in the higher vibrational frequency of Zr1 −xHfxCo and the larger reduced mass will bring about the lower vibrational frequency. For Zr1−xHfxCo, the influence of interaction between Zr1−xHfx and Co on vibrational frequency may be more important than the reduced mass.

3.4. Thermodynamic properties

Knowledge of the thermodynamic properties is essential for studying crystal stability and chemical reactivity, but there are no publications focusing on thermodynamic properties of Zr1 −xHfxCo except the ZrCo. As is well known, the first-principles for phonon calculations are limited to T = 0 K yet the detailed thermodynamic properties including internal energy E, entropy S, constant-volume specific heat CV per unit cell of the crystals could be derived by phonons.[42]

In order to ensure the accuracies of thermodynamic properties for Zr1−xHfxCo with the VCA method, here we perform an ordered SC calculation of Zr0.5Hf0.5Co. The thermodynamic properties for Zr0.5Hf0.5Co are obtained by the VCA and the SC methods and listed in Table 3. The ZrCo supercell is built with the 2×1×2 of a unit cell, there are two of Zr positions substituted for by elements Hf as shown in Fig. 5. The supercell contains two Zr, two Hf and four Co atoms, and its constituent can be reduced to the formula Zr0.5Hf0.5Co. As can be seen from Table 3, the calculated thermodynamic results obtained with the two methods are almost equal and the maximum error is less than 0.3%. Hence, we believe that the VCA method is a reliable method for predicting the thermodynamic properties of the ternary Zr1−xHfxCo alloy.

Fig. 5. (color online) Crystal structure of Zr0.5Hf0.5Co.
Table 3.

Calculated results of the internal energy E , entropy S , and constant-volume specific heat CV per unit cell for Zr1−xHfxCo at different values of temperature T (K), obtained with the VCA method and SC method.

.

Gibbs free energy G is another important thermodynamic quantity. In the hydriding/dehydriding reaction the change of G can be used to determine the direction of the reaction. As is well known, the G can be obtained from the following formula:

where T and S are the temperature and entropy, respectively. H is the enthalpy, which can be expressed as
so
for solid and liquid phase, the value of PV at atmospheric pressure is very small compared with that of Gibbs free energy G, so PV can be ignored. Then, the G can be expressed as
The calculated values of E, S, CV, and G at temperatures from 0 K to 1000 K are illustrated in Figs. 6(a)–6(d), respectively (the corresponding data for E, S, CV, and G are provided in the supplementary materials from Table S1 to Table S4).

Fig. 6. (color online) Thermodynamic properties for Zr1−xHfxCo: (a) internal energy E, (b) entropy S, (c) specific heat CV, and (d) Gibbs free energy G.

From Fig. 6, one can see that the differences among these thermodynamic quantities for Zr1−xHfxCo are very small, whereas the further analysis reveals that the increase of Hf content can cause the thermodynamic quantities to have some changes. In detail, as the Hf content increases, the calculated value of the internal energy E and Gibbs free energy G decrease, while the entropy S and constant-volume specific heat CV increase in the selected temperature ranges.

Furthermore, in order to compare our calculated results with other reported theoretical results[16,19] and experimental[19] data for ZrCo, the calculated values of E, S, and CV at selected temperatures ranging from 0 K to 1000 K for ZrCo are listed in Table 4. At a constant pressure, the change of enthalpy can be expressed as , where is very small compared with in the solid phase, as a result in our work . As can be seen from Table 4, our results are somewhat less than the experimental data reported by Chattaraj et al., and the deviation increases with temperature increasing. The following reasons may cause the discrepancy: (i) in our calculation, the electronic contribution to the entropy S is neglected, and only the vibration entropy is considered;[43] (ii) our thermodynamic calculations are based on the harmonic approximation which has been successfully used to calculate the thermodynamic functions in most cases,[16,18,43–46] and the anharmonicity is ignored, whereas the effect of anharmonicity on thermodynamic function may be important and increased with temperature increasing;[43] (iii) the enthalpy increments in a temperature range from 642 K–1497 K are measured by using a high temperature inverse drop calorimeter and the following thermodynamic functions are derived from the measured enthalpy increments at 0.1 MPa,[19] but our calculations are based on an ideal single crystal under the 0 Pa. In comparison with the theoretical results, our calculated results of thermodynamic function are reasonably consistent with the results obtained from the PWSCF package and VASP code by Li et al.[16] and Chattaraj et al.[19] respectively. Accordingly, our predicted thermodynamic properties for Zr1−xHfxCo are reliable.

Table 4.

Calculated and experimental data of E , S , (kJ/mol), (kJ/mol) per unit cell of ZrCo.

.
4. Conclusions

In summary, we investigate the structural, electronic, vibrational, and thermodynamic properties of Zr1−xHfxCo (x is the concentration of constituent element Hf, and changes from 0 to 1) using a first principles approach and the VCA method.

In our work, the equilibrium structures of Zr1−xHfxCo are investigated and reveal that as the Hf concentration x gradually increases from 0.0 to 1.0, the lattice constant reduces from 3.217 Å to 3.195 Å very slowly. In general, the obtained electronic structures of Zr1−xHfxCo exhibit similar behaviors, but further analysis indicates that the metallic nature is enhanced and the electrical conductivity turns better with the increase of Hf content. Moreover, as Hf concentration increases from 0 to 1, the Fermi energy gradually increases from −6.96 eV to −6.21 eV, and the decreases from 2.795 electrons/eV f.u. to 2.594 electrons/eV f.u., both of which imply the decrease of chemical stability.

The dynamic properties with different Hf content in Zr1 −xHfxCo compound are studied within DFPT, and the results show that the increase of Hf content makes the optical modes softened, the acoustical modes essentially unchanged, and the PHDOS significantly moves towards a low wave number generally. The maximum vibrational frequency decreases gradually from about 223 cm−1 to 186 cm−1, which implies that there appear the lower scattering gradient and lower phonon group velocity with the decrease of the highest vibrational frequency. Finally, the complete thermodynamic data including internal energy E, constant-volume specific heat CV, entropy S, and Gibbs free energy G are obtained and analyzed. The results demonstrate that there are small differences in the thermodynamic property with the increase of the Hf content, and further analysis suggests that as the Hf content increases, the calculated values of the internal energy E and Gibbs free energy G decrease, while the entropy S and constant-volume specific heat CV increase in the selected temperature ranges.

We expect some of our calculated results, such as thermodynamic quantities will provide useful information about tritium storage and transport.

Appendix A: Supplementary material

The data for parameters E, S, C, and G are listed in the following Tables S1–S4.

Table S1.

The calculated result of internal energy E per unit cell for Zr1−xHfxCo at different temperature T (K), x represents the concentration of constituent element Hf.

.
Table S2.

The calculated result of entropy S (J/mol.K) per unit cell for Zr1−xHfxCo at different temperature T(K), x represents the concentration of constituent element Hf.

.
Table S3.

The calculated result of constant-volume specific heat CV per unit cell for Zr1−xHfxCo at different temperature T (K), x represents the concentration of constituent element Hf.

.
Table S4.

The calculated result of Gibbs free energy G per unit cell for Zr1−xHfxCo at different temperature T (K), x represents the concentration of constituent element Hf.

.
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