CHIN.PHYS.LETT.
Vol. 25, No. 11 (2008) 4086
Structural, Elastic and Electronic Properties of ReO2
*
LI Yan-Ling(李延龄)1,2,3** , ZENG Zhi(曾雉)2*** 1
2
Department of Physics, Xuzhou Normal University, Xuzhou 221116 Key Laboratory of Materials Physics, Institute of Solid State Physics, Chinese Academy of Sciences, Hefei 230031 3 Graduate School of the Chinese Academy of Sciences, Beijing 100049
(Received 3 June 2008) Structural, elastic and electronic properties of ReO2 are investigated by first-principles calculations based on density functional theory. The ground state of ReO2 has an orthorhombic symmetry which belongs to space group 𝑃 𝑏𝑐𝑛 with 𝑎 = 4.7868 ˚ A 𝑏 = 5.5736 ˚ A and 𝑐 = 4.5322 ˚ A. The calculated bulk moduli are 322 GPa, 353 GPa, and 345 GPa for orthorhombic, tetragonal, and monoclinic ReO2 , respectively, indicating that ReO2 has a strong incompressibility. ReO2 is a metal ductile solid and presents large elastic anisotropy. The obtained Debye temperatures are 850 K for orthorhombic, 785 K for tetragonal, and 791 K for monoclinic ReO2 .
PACS: 62. 20. −x, 71. 15. Mb,71. 20. −b Transitional-metal dioxide (TMO2 ) compounds have been very extensively investigated over decades due to their interesting fundamental properties.[1−10] The rich physical properties of TMO2 s are associated with their electronic structures and atomic arrangements. The hybridization of TM-𝑑 and O-𝑝 electrons contribute to their peculiar electronic properties, yielding the strong covalence bonds between TM and O atoms which make them possess high bulk moduli. Some of TMO2 ’s present the tetragonal (TiO2 type rutile) structures, such as TaO2 , NbO2 , RuO2 , RhO2 , OsO2 , and IrO2 and the others prefer monoclinic structures, such as MoO2 , TcO2 , and WO2 .[1] In the case of ReO2 , the structural form is of monoclinic MoO2 (space group 𝑃 21 /𝑐, 𝑍 = 4, referred to 𝛼-ReO2 ) below about 300∘ C. While above 300∘ C, it transforms to an orthorhombic form (space group 𝑃 𝑏𝑐𝑛, 𝑍 = 4, referred to 𝛽-ReO2 ) with a structure characterized by zigzag chains of Re atoms propagating along the 𝑐 axis of the unit cell.[11] Recently, rutile-type structure has been reported for ReO2 (space group 𝑃 42/𝑚𝑛𝑚, referred to rutile-ReO2 ) experimentally.[12] So far, for ReO2 , the investigations are most focused on studying its catalyst, electrochemistry and technical application as a protective material. The fundamental researches are still needed for ReO2 . Rogers et al.[1] determined the temperature dependence of electrical resistivity, exhibiting that ReO2 is a typical metal-like conductor. Corrˆea et al.[13] refined the monoclinic ReO2 structure from XRD by Rietveld method, pointing out the monoclinic ReO2 with lattice parameters 𝑎 = 5.615(3) ˚ A 𝑏 = 4.782(2) ˚ A 0 ˚ 𝑐 = 5.574(2) A 𝛽 = 120.13 . Ivanovskii et al.[13] calculated the electronic property of rutile-ReO2 by us-
ing projected augmented wave method. In this Letter, the structural, elastic and electronic properties of ReO2 are investigated by means of density functional theory (DFT) within the framework of local density approximation (LDA). To our knowledge, the elastic property of ReO2 has not been reported so far. All calculations are performed by the CASTEP code[14] using ab initio pseudopotentials based on DFT within the LDA with the exchange-correlation functional of Ceperley and Alder as parametrized by Perdew and Zunger (LDA-CAPZ). All the possible structures are optimized by the BFGS algorithm which provides a fast way of finding the lowest energy structure and supports cell optimization. In the calculation, the interaction between the ions and the electrons is described by using Vanderbilt’s supersoft pseudopotential with the cutoff energy of 380 eV. The Monkhorst-Pack 𝑘-point grid with a fine quality Brillouin zone sampling of 2𝜋 ×0.04 ˚ A−1 is used in the calculations. In the geometrical optimization, all forces on atoms are converged to less than 0.002 eV/˚ A, all the stress components are less than 0.02 GPa, and the tolerance in self-consistent field (SCF) calculation is 5.0 × 10−7 eV/atom. Relaxation of the internal degrees of freedom is allowed at each unit cell compression or expansion. The elastic constants of ReO2 are obtained by using the finite strain technique. From the full elastic constant tensor we determine the bulk modulus 𝐵 and the shear modulus 𝐺 according to the Voigt–Reuss–Hill (VRH) approximation.[15] Young’s modulus 𝐸 and Poisson’s ratio 𝜈 can be calculated by the formula 𝐸=
9𝐵𝐺 , 3𝐵 + 𝐺
𝜈=
3𝐵 − 2𝐺 . 2(3𝐵 + 𝐺)
* Supported by the National Science Foundation of China under Grant Nos 10504036 and 90503005, the special Funds for Major State Basic Research Project of China under Grant No 2007CB925004, the Knowledge Innovation Programme of Chinese Academy of Sciences, and Director Grants of CASHIPS. Part of the calculations were performed in the Shanghai Supercomputer Center. ** Email:
[email protected] *** Email:
[email protected] c 2008 Chinese Physical Society and IOP Publishing Ltd ○
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Table 1. Equilibrium lattice parameters, 𝑉0 (˚ A3 ), 𝑎 (˚ A), 𝑏 (˚ A), 𝑐 (˚ A), density 𝜌 (g/cm3 ), bulk modulus 𝐵0 and its pressure derivative 𝐵0′ , relative total energy 𝐸tot (eV). 𝑉0 and 𝐸tot are of per ReO2 molecule. 𝛽-ReO2
𝑉0 30.2287
Rutile-ReO2
31.1548
𝛼-ReO2
31.1859
𝑎 4.7868 4.81 4.8471 4.7983 5.3254 5.615
𝑏 5.5736 5.64
𝑐 4.5322 4.60 2.6521 2.8077 5.5283 5.574
4.8345 4.782
𝜌 11.7271
𝐵0 321.8
𝐵0′ 4.242
𝐸tot 0
11.3785
345.7
4.587
0.402
11.3671
344.6
4.473
0.413
Ref. This [11] This [12] This [13]
Table 2. Zero-pressure elastic constants 𝑐𝑖𝑗 (GPa), the isotropic bulk modulus 𝐵 (GPa), shear modulus 𝐺 (GPa), Young’s modulus 𝐸 (GPa) and Poisson’s ratio 𝜈. 𝛽-ReO2 Rutile-ReO2 𝛼-ReO2
𝑐11 414 484 517
𝑐22 521 471
𝑐33 565 501 501
𝑐44 165 149 314
𝑐55 231 231 135
𝑐66 297 376 208
𝑐12 306 391 264
Fig. 1. Total energy per ReN molecule as a function of volume (EOS).
The optimized equilibrium structural volume per molecule, lattice constants, density, relative total energy per molecule, the bulk moduli and their pressure derivatives for three phases of ReO2 , i.e., 𝛽-ReO2 , rutile-ReO2 , and 𝛼-ReO2 , are displayed in Table 1. Here the equilibrium volume, bulk moduli and their pressure derivatives are obtained by fitting the total energy as a function of volume to the third-order Birch-Murnaghan equation of states (EOS).[16] The total energy as a function of volume are plotted in Fig. 1. From Table 1, one can see that the calculated equilibrium parameters agree well with the experimental values. The calculations of total energy for three phases at zero pressure, together with Fig. 1 show that the ground state of ReO2 is 𝛽-ReO2 , in which four rhenium atoms occupy the 4𝑐 Wychoff site (0, 0.1082, 0.25) and eight oxide atoms hold 8𝑑 Wychoff site (0.2416, 0.3592, 0.0914). It is worthy to note that there are only slight differences in equilibrium volume, density, and total energy between rutileReO2 and 𝛼-ReO2 , which may results in their similar electronic property discussed later. For rutile-ReO2 , two transition metal atoms lie in the 2𝑎 Wychoff site and four oxygen atoms occupy the 4𝑓 Wychoff site
𝑐13 213 237 214
𝑐23 187 337
𝑐15 3.4
𝑐25 65
𝑐35 −17
𝑐46 95
𝐵 322 353 345
𝐺 173 144 147
𝐸 441 381 385
𝜈 0.27 0.32 0.31
(0.2816, 0.2816, 0). For 𝛼-ReO2 , four rhenium atoms occupy 2𝑎 and 2𝑒 sites and eight oxygen atoms hold 4𝑒 (0.3584, 0.2197, 0.2169) and 4𝑒 (0.1415, 0.7197, 0.2831). The optimized angle 𝛽, internal parameter, is 118.8∘ in 𝛼-ReO2 . The bulk moduli obtained are 321.8 GPa, 345.7 GPa, and 344.6 GPa for 𝛽-ReO2 , rutile-ReO2 , and 𝛼-ReO2 , respectively, which are comparable with that (332 GPa) of orthorhombic ZrO2 [17] as well as that (340 GPa) of HfO2 ,[18] implying that ReO2 is a strong incompressible solid. In order to illustrate the compressibility of ReO2 quantitatively, we observe the changes in length of crystallographic axes 𝑎, 𝑏 and 𝑐 when exerting the static pressure on ReO2 . That is, we calculate the 𝑐/𝑎, 𝑏/𝑎, and 𝑏/𝑐 ratios as a function of pressure. The results show that all 𝑐/𝑎, 𝑏/𝑎, and 𝑏/𝑐 ratios increase with the increasing pressure for 𝛼- and 𝛽-ReO2 , whereas for rutile-ReO2 the ratio 𝑐/𝑎 decreases with the increasing pressure. Hence, one can come to a conclusion that 𝑏-axis is the most difficult to be compressed, while 𝑐-axis is the second, and 𝑎-axis is the last for 𝛼- and 𝛽-ReO2 , and the situation however is the reverse for the 𝑐/𝑎 ratio of rutile-ReO2 . The difference in compressibility along different directions originates from the different electronic repulsion (i.e., electron distribution) in nature. Moreover, we notice that the angle 𝛽 in 𝛼-ReO2 decreases with the increasing pressure. The elastic constants are shown in Table 2. The number of independent elastic constants decreases as the symmetry of the crystalline system increases and it reduces to six, nine and thirteen for tetragonal, orthorhombic and monoclinic structures, respectively. Our calculated elastic constants 𝑐𝑖𝑗 satisfy the Born– Huang criteria for stability, suggesting that the three phases of ReO2 are mechanically stable. Using the calculated elastic constants, we calculate bulk modulus 𝐵 and shear modulus 𝐺 (Table 2) according to VRH approximation. The obtained bulk moduli are 322 GPa for 𝛽-ReO2 , 353 GPa for rutile-ReO2 , and 345 GPa for
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𝛼-ReO2 , which agree well with the ones by fitting the total energy as a function of volume to the third-order Birch–Murnaghan EOS mentioned above, providing a consistent estimation of the compressibility for ReO2 . Then we calculate Young’s modulus 𝐸 and Poisson’s ratio 𝜈 considering their importance for technological and engineering applications. In general, Poisson’s ratio reflects the stability of a crystal against shear. All the calculated Poisson’s ratios are larger than 0.25, which means that strong elastic anisotropy in ReO2 .[19] The ratio of the bulk modulus to shear modulus 𝐵/𝐺 are 1.86, 2.45, 2.35 for 𝛽-, rutile-, and 𝛼- ReO2 , respectively, indicating that ReO2 is a ductile solid.[20]
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the contribution of O-𝑝 and Re-𝑑 orbitals. Thirdly, the DOSs of Re-𝑑 and O-𝑝 are energetically degenerate from the bottom of the valence band to the top of conduction band, indicating the possibility of covalent bonding between Re and O atoms, which contributes to high bulk modulus. Moreover, the total DOS distribution around Fermi level is near flat for 𝛽-ReO2 , whereas the total DOS around Fermi level have a peak for rutile-ReO2 and 𝛼-ReO2 . It confirms the total energy calculations: the 𝛽-ReO2 phase is the most stable among those phases discussed.
Table 3. Bulk modulus along the crystallographic axes 𝑎, 𝑏, and 𝑐 (𝐵𝑎 , 𝐵𝑏 , and 𝐵𝑐 ), percent elastic anisotropy for shear and bulk moduli 𝐴𝐺 (in %), 𝐴𝐵 (in %), and longitudinal, transverse, average elastic wave velocity (𝜐𝑙 , 𝜐𝑡 , and 𝜐𝑚 in m/s) as well as the Debye temperature Θ𝐷 (K) at the theoretical equilibrium volume. 𝐵𝑎 𝐵𝑏 𝛽-ReO2 758 1282 Rutile-ReO2 1234 𝛼-ReO2 888 1276
𝐵𝑐 𝐴𝐵 985 0.367 814 1 1011 0.439
𝐴𝐺 10.58 21.44 20.21
𝜐𝑙 6868 6923 6896
𝜐𝑡 3843 3560 3592
𝜐𝑚 4277 3987 4019
Θ𝐷 850 785 791
According to the obtained elastic constants, we therefore discuss the elastic anisotropy of ReO2 . We calculate the directional bulk modulus along crystallographic axes 𝑎, 𝑏, and 𝑐 (𝐵𝑎 , 𝐵𝑏 , and 𝐵𝑐 ) and the percent elastic anisotropy (Table 3). The obtained directional bulk modulus values again confirm the difference in compressibility along different directions as the discussed above. In Table 3, 𝐴𝐵 and 𝐴𝐺 represent the anisotropy in compressibility and in shear. A value of zero represents elastic isotropy and a value of 1 (100%) is the largest possible anisotropy. Obviously, the ReO2 possesses strong shear anisotropy and weak bulk anisotropy. Further, we calculate the Debye temperature Θ𝐷 , considering that it correlates with many physical properties of materials, such as specific heat, elastic constants, and melting temperature. Debye temperature can be estimated from average sound velocity 𝜐𝑚 and molecular weight together with density 𝜌.[19] The longitudinal, transverse, average elastic wave velocity (𝜐𝑙 , 𝜐𝑡 , and 𝜐𝑚 , respectively) and Debye temperature Θ𝐷 are also listed in Table 3, in which sound velocities are derived from the calculated bulk modulus 𝐵, shear modulus 𝐺, and density 𝜌.[19] The calculated Debye temperatures are 850 K, 785 K, 791 K for orthorhombic, tetragonal, and monoclinic ReO2 , respectively. The total and partial density of states (DOS) of ReO2 near the Fermi level are shown in Fig. 2. Firstly, dominant contributions to DOS at Fermi level (𝑁 (𝐸𝐹 )) come from Re-𝑑. The large 𝑁 (𝐸𝐹 ) value of ReO2 unfolds its metallicity. Secondly, valence or conduction bands near Fermi level are mainly from
Fig. 2. Partial and total DOS with respect to the Fermi level for the 𝛼-ReO2 , 𝛽-ReO2 , and rutile-ReO2 .
In summary, the structural, elastic and electronic properties of ReO2 are investigated by means of firstprinciples total energy calculations. Three phases, i.e., orthorhombic 𝛽-ReO2 , tetragonal rutile-ReO2 , and monoclinic 𝛼-ReO2 , are systematically discussed. The results show that the ground state of ReO2 is 𝛽ReO2 with 𝑃 𝑏𝑐𝑛 symmetry. All phases discussed here are stable mechanically and have strong incompressibility because of their high bulk moduli (322 GPa, 353 GPa, and 345 GPa for 𝛽-ReO2 , rutile-ReO2 , and 𝛼-ReO2 , respectively). ReO2 is a metal ductile solid and has large elastic anisotropy. The Debye temperatures are 850 K for orthorhombic, 785 K for tetragonal,
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and 791 K for monoclinic ReO2 .
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