J Supercond Nov Magn (2010) 23: 717–719 DOI 10.1007/s10948-010-0675-2
O R I G I N A L PA P E R
Superconductivity in Hydrogen-rich Material: GeH4 Chao Zhang · Xiao-Jia Chen · Yan-Ling Li · Viktor V. Struzhkin · Russell J. Hemley · Ho-Kwang Mao · Rui-Qin Zhang · Hai-Qing Lin
Received: 9 December 2009 / Accepted: 7 January 2010 / Published online: 21 January 2010 © Springer Science+Business Media, LLC 2010
Abstract The electronic properties, lattice dynamics, and electron–phonon coupling of the Cmmm phase of GeH4 have been studied by first-principle calculations using density functional perturbation theory. The electronic band structure shows the Cmmm phase metallic nature. It is found strong electron phonon interaction, and the superconducting critical temperature, predicted by Allen–Dynes modified McMillan equation, is about 40 K at 20 GPa. Keywords First-principle calculations · Electron–phonon coupling · Superconducting critical temperature
1 Introduction Metallic hydrogen, a leading candidate for clean energy carriers in future transportation applications, also serves as one of potential candidates for high-temperature superconductivity [1]. However, hydrogen remains insulating at extremely high pressure, at least up to ∼342 GPa [2]. To cirC. Zhang · R.-Q. Zhang Department of Physics and Materials Science, City University of Hong Kong, Hong Kong, China C. Zhang · Y.-L. Li · H.-Q. Lin () Department of Physics and Institute of Theoretical Physics, Chinese University of Hong Kong, Hong Kong, China e-mail:
[email protected] X.-J. Chen · V.V. Struzhkin · R.J. Hemley · H.-K. Mao Geophysical Laboratory, Carnegie Institution of Washington, Washington, DC 20015, USA X.-J. Chen Department of Physics, South China University of Technology, Guangzhou 510640, China
cumvent this problem, it is proposed that Group IVa hydrides would become metallic at relatively lower pressures and exhibit superconductivity [3]. This suggestion has motivated considerable theoretical and experimental activities [4–10]. As a heavy Group IVa hydride, GeH4 has been extensively investigated [11, 12]. Under pressure, GeH4 undergoes a series of phase transitions. At lower pressure, it stabilizes in P 21 /c phase, and then transforms to Cmmm at 15 GPa, and then to two other phases. Since the Cmmm phase is a lower pressure phase of GeH4 , we focus on this structure and discuss its electronic, lattice dynamical, and possible superconducting properties in this paper.
2 Computational Details The calculations have been performed using the firstprinciples pseudopotential plane-wave method based on the density functional perturbation theory [13] through the Quantum–Espresso package [14]. The generalized gradient approximation (GGA) exchange correlation functional within the Perdew–Burke–Ernzerh (PBE) [15] of parameterization was employed. A Troullier–Martins normconserving [16] scheme was used to generate pseudopotential. The energy cutoff of 60 Ry was used for the plane wave basis. Monkhorst–Pack (MP) [17] meshes were used for Brillouin zone (BZ) integrations in the electronic calculations (k mesh), and the BZ sampling in phonon and electron phonon coupling (EPC) calculations (q mesh). The EPC matrix elements have been calculated in the first BZ on 3 × 3 × 3 q mesh obtained with 18 × 18 × 18 k mesh employing Gaussian smearing of 0.035 Ry.
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Fig. 1 Electronic properties of the Cmmm GeH4 phase at the pressure of 20 GPa: (a) electronic band structure, (b) density of states (DOS)
3 Results and Discussion The Cmmm phase of GeH4 is a C-centered orthorhombic structure in which the conventional cell contains two GeH4 units. The Ge atoms occupy Wyckoff 2c position, and two inequivalent H atoms, named by H1 and H2, occupy Wyckoff 4i and 4f positions, respectively. The calculated electronic properties of Cmmm phase at 20 GPa are shown in Fig. 1, including electronic band structure and density of states (DOS). Figure 1(a) displays the electronic band structure along the principal symmetry directions of BZ. Clearly, it reveals the metallic properties of Cmmm phase. The large dispersion bands cross the Fermi level provides several electron and hole pockets at Fermi level. Moreover, an interesting feature is the extended flat band around the Fermi level from Γ to S and S to R point in the first BZ. The simultaneous occurrence of the flat band near the Fermi level is essential to superconductivity. The calculated DOS, depicted in Fig. 1(b), shows a strong hybridization between Ge and H atoms in a large energy range. In the energy range from −3 eV to 5 eV, the DOS of Ge atoms is about two times of that of H atoms. The phonon band structure and projected phonon DOS of Cmmm phase of GeH4 at 20 GPa are shown in Figs. 2(a) and 2(b). The absence of imaginary vibration modes shows that the Cmmm phase is stable. Overall, the heavy Ge atoms dominate the low-frequency vibrations, the light H atoms contribute significantly to the high-frequency modes. It can be seen from Fig. 2(b) that the H1 vibrational modes
from 200 to 900 cm−1 and around 4150 cm−1 , while the H2 vibrational modes dominate the frequency from 500 to 1500 cm−1 . The possibility of superconductivity for the Cmmm structure of GeH4 is discussed by using the modified McMillan equation by Allen and Dynes, ωlog 1.04(1 + λ) Tc = exp − , 1.2 λ − μ∗ (1 + 0.62λ) where ωlog is the logarithmic average of phonon frequencies and μ∗ is the Coulomb pseudopotential representing Coulombic repulsion. The calculated spectral function α 2 F (ω) and the integrated λ(ω) as a function of frequency are shown in Fig. 2(c). The obtained electron–phonon coupling parameter λ(ω) is 1.03 at 20 GPa, indicating a rather strong electron–phonon coupling in solid GeH4 . The lowfrequencies below 400 cm−1 , which are from Ge and H1 vibrations, contribute 15% of the total λ(ω) parameter. The H1 and H2 vibrational modes between 400 and 700 cm−1 contribute 30% to the total λ(ω) parameter. The H2 vibrational modes from 700 to 1650 cm−1 contribute 50% to the total λ(ω) parameter. The remaining contribution comes from H1 around 4150 cm−1 . Based on the obtained α 2 F (ω) and λ(ω), we estimated the superconducting transition temperature Tc of 47 and 36 K by using the Coulomb pseudopotential of 0.1 and 0.15, respectively. This relatively high transition temperature is a result of strong electron–phonon coupling in the Cmmm GeH4 .
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Fig. 2 (a) The phonon band structure and (b) projected phonon density of states (PhDOS) projected on Ge, H1, and H2 atoms of the Cmmm GeH4 at 20 GPa. (c) The Eliashberg phonon spectral function α 2 F (ω) and the electron–phonon integral λ(ω)
4 Conclusions The electronic properties, lattice dynamics, and electron– phonon coupling of Cmmm phase of GeH4 at 20 GPa have been studied using density functional perturbation theory. The electronic band structure reveals the metallic character of Cmmm phase. The calculated EPC parameter λ is 1.03 at 20 GPa, indicating strong electron phonon interaction. The corresponding critical temperature Tc is about 40 K, which means GeH4 at low pressure is a good superconducting candidate. Acknowledgements We are grateful to J.A. Montoya, L.L. Sun, and Z.X. Zhao for discussions and comments. This work was supported by the HKRGC (402108) and NSFC (10874046); the US DOE-BES (DEFG02-02ER45955), DOE-NNSA (DEFC03-03NA00144), and NSF (DMR-0205899).
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