Structural, Electronic, Dynamical, and Superconducting Properties in

Article pubs.acs.org/JPCC

Structural, Electronic, Dynamical, and Superconducting Properties in Dense GeH4(H2)2 Guohua Zhong,*,† Chao Zhang,‡ Xiaojia Chen,§ Yanling Li,∥ Ruiqin Zhang,⊥ and Haiqing Lin‡,⊗,† †

Center for Photovoltaics and Solar Energy, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences, Shenzhen 518055, China ‡ Beijing Computational Science Research Center, Beijing 100089, China § Department of Physics, South China University of Technology, Guangzhou 510640, China ∥ Department of Physics, Xuzhou Normal University, Xuzhou 221116, P. R. China ⊥ Department of Physics and Materials Science, City University of Hong Kong, Hong Kong, China ⊗ Department of Physics, Institute of Theoretical Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China ABSTRACT: Hydrogen-rich materials have fascinating physical and chemical properties such as various structures and superconductivity under high-pressure. In this study, structural, electronic, dynamical, and superconducting properties of GeH4(H2)2 are investigated based on the first-principles calculations. We first predict several phase transitions of GeH4(H2)2 under pressure. Below 28 GPa, two degenerated structures with I4̅m2 and Pmn21 symmetries are preferred, which can be viewed as the distortion of the experimentally observed fcc structure. Then, the GeH4(H2)2, via a triclinic phase that stabilizes in the pressure range of 28−48 GPa, transforms into a metallic orthorhombic phase in which appears the metallization induced by pressure. Another metallic phase with P21/c symmetry enters the phase diagram at around 220 GPa, which is more stable than the case of a decomposed material, and its stability is also confirmed by including the zero point energy correction. In the high-pressure P21/c phase, the superconductivity is found, and the superconducting transition temperature is predicted to be as high as 76−90 K at 250 GPa. This superconductivity mainly results from the local vibrations of more H2 units, though the vibration of Ge in an H2-formed grid also contributes to the electron−phonon interaction. This study is helpful for understanding the superconducting mechanism on hydrogen-rich compounds.

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INTRODUCTION Since Wigner and Huntington suggested that hydrogen will transform to metal from insulator at sufficiently high pressure in 1935,1 hydrogen under pressure has attracted much attention from the scientific community. This interest was further fueled by Ashcroft’s prediction in 19682 that the metallic hydrogen might even be a high-temperature superconductor. This suggestion has motivated considerable experimental and theoretical activities. However, hydrogen remains insulating at extremely high pressure, at least up to 320 GPa.3 Although estimations based on the infrared data3 revealed that the band gap closure for solid hydrogen should occur at about 450 GPa, the pressure was extremely difficult to be achieved by the diamond-anvil-cell technique. As an alternative, Ashcroft proposed4 that the hydrogen-rich alloys should transform into metal under the relatively lower pressure by means of chemical precompressions from the comparable weight elements. Therefore, hydrogen-rich group IVA hydrides (i.e., CH4,5−10 SiH4,11−22 GeH4,23−27 SnH4,28−32 and PbH433) and alkaline earth hydrides34−43 have been extensively explored. Inspiringly, recent experiments16,17 proved that the metallization in solid SiH4 takes place at about 60 GPa. Moreover, the super© 2012 American Chemical Society

conductivity with a transition temperature (Tc) of 17 K at 96 and 120 GPa was revealed in silane,17 though debates remained.20 To understand the mechanism of superconductivity, high-pressure structures have been examined in detail, as well as their evolution with pressure. It was found that silane15,19 basically remained at its molecular form up to 220 GPa with the closest H−H distance of 1.35 Å. The distance is far from the dimer length (0.74 Å) while being very similar to the Si−H distance of 1.475 Å. However, the semimolecular H2 units with H−H lengths of 0.87 Å (220 GPa) and 0.79 Å (120 GPa) were observed in the high-pressure structures of GeH4 and SnH4. The higher-temperature superconductivity with Tc ≈ 64 K at 220 GPa for germane25,27 and ∼62 K at 200 GPa for stannane31 have been theoretically predicted, respectively. As a result, these H2 units have been found to contribute significantly to the superconductivity, which will inevitably motivate further seeking of high-temperature superconductors Received: November 16, 2011 Revised: January 8, 2012 Published: January 31, 2012 5225

dx.doi.org/10.1021/jp211051r | J. Phys. Chem. C 2012, 116, 5225−5234

The Journal of Physical Chemistry C

Article

them to other pressure points. All structural optimizations were performed using the Perdew−Burke−Ernzerhof generalized gradient approximation (GGA)61 density functional theory and projector augmented wave pseudopotentials62 as implemented in the Vienna ab initio simulation package (VASP).63 An energy cutoff of 800 eV was used for the plane wave basis sets, and 16 × 16 × 16, 8 × 8 × 8, and 6 × 6 × 6 Monkhorst−Pack k-point grids were used for Brillouin zone (BZ) sampling of two, three, and four GeH4(H2)2 molecular cells, respectively. In the geometrical optimization, all forces on atoms were converged to less than 0.001 eV/Å, and the total stress tensor was reduced to the order of 0.01 GPa. The lattice dynamical and superconducting properties were calculated by the QUANTUM ESPRESSO package (QE)64 with a cutoff energy of 60 and 450 Ry for wavefunctions and charge densities, respectively. Forces and stresses for the converged structures were optimized and checked to be within the error between the VASP and QE code. The Troullier−Martin norm-conserving scheme65 was used to generate the pseudopotentials for Ge and H. A 16 × 16 × 12 Monkhorst-Pack k-point grid with Gaussian smearing of 0.03 Ry was used for the phonon calculations at 4 × 4 × 3 q-point mesh, and a double k-point grid was used in the calculation of the electron−phonon interaction matrix element for the P21/c phase of GeH4(H2)2. For selective cases, we also compared the GGA results with those from the Ceperley−Alder local density approximation (LDA)66 as parametrized by Perdew and Zunger,67 the hybrid functional calculation (HSE06),68−70 and GW approximation (GWA).71,72 For low-pressure I4m ̅ 2 phase, at 8.5 GPa, the lattice constants viewed as distorted fcc structure are a′ = 6.349 and c′ = 6.248 Å obtained from GGA, which is only 1−2% less than that from experiment (a = 6.411). Furthermore, the results from LDA (a′ = 6.116 and c′ = 6.021 Å) are 3% less than those from GGA. For high-pressure P21/c phase at 250 GPa, however, the crystal lattice constants (a = 2.902, b = 2.859, and c = 5.884 Å) from LDA are almost consistent with those from GGA as shown in Table 1. In other words, at high pressure, the obtained results of GeH4(H2)2 are consist by GGA and LDA. As the LDA/GGA underestimates the band gaps, we have also performed the GWA and hybrid functional calculations. We found that the band gap of I4m ̅ 2 phase at 8.5 GPa were 4.6, 5.5, and 5.8 eV from GGA, GWA, and HSE06 methods, respectively. The band gap strongly depends on the adopted methods. It is well-known that GWA is much better than LDA/ GGA when predicting band gaps. Moreover, Ramzan et al.73 and Kaewmaraya et al.74 have proved that GGA underestimated the metallization pressure comparing with GWA. Unfortunately, these metallization pressures predicted from GGA and GWA are both larger than experiments values, which have been explained by Ramzan et al.73 in the case of SiH4(H2)2. However, the difference among several methods is little for high-pressure metallic phase. Considering the balance between precision and calculation loads, therefore, all the results and discussion presented in next section are based on the GGA functional.

and studying of the superconductivity in these materials containing H2 units. Recently, researchers have devoted themselves to studying the H2-containing compounds by way of the hydrogen-rich closed-shell group IVA hydrides (CH4, SiH4, GeH4, and SnH4) absorbing additional H2 molecules at high pressures, such as CH4(H2)2, (CH4)2H2, CH4(H2)4, CH4H2,44 SiH4(H2)2,45,46 and GeH4(H2)2.47 In addition, some H2-containing molecular compounds have been also obtained by the H2 interacting with Ar,48 H2O,49,50 NH3BH3,51,52 and Xe.53 For closed-shell nonpolar molecules, packing rules analogous to the empirical Hume−Rothery rules for metallic alloys have been invoked to explain compound formation.48,53,54 However, in real molecular systems, the driving force for compound formation may be more complex due to a combination of interactions. It was found that these novel closed-shell nonpolar molecules, SiH4(H2)245,46 and GeH4(H2)2,47 formed the high-pressure molecular solids with unusually strong intermolecular interaction, but Raman and infrared spectroscopic measurements showed that multiple H2 vibrations substantially softened from bulk solid hydrogen and the frequencies of several Raman and infrared H2 vibrations decreased with the increase of pressure, indicating anomalous attractive interaction in these two closedshell nonpolar molecules.45−47 This behavior is different from those in other simple molecular compounds where the repulsion dominates intermolecular interaction. For SiH4(H2)2, theoretical studies proposed a distorted facecentered cubic (fcc) lattice with orientationally disordered H2 units under relatively low pressure.55 The dynamical behavior of the H2 molecules was also established in high-pressure structures of SiH4(H2)2, with the rotating of H2 molecules, whereas the silane molecules remained rigid.56 Moreover, other molecular dynamics calculations57 suggested a structure with orientationally disordered silane and hydrogen with their centers of mass arranged in SiH4(H2)2. The intermolecular interaction between H2 and SiH4 molecules results in the H−H bond in H2 units weakening, and the covalent bonding interaction strengthens in SiH4(H2)2 with the increase of pressure.56−60 Furthermore, SiH4(H2)2 was predicted to become metallic near 120 GPa for P1 structure,56 near 92 GPa for Cc structure,58 and near 200 GPa for F43̅ m structure.59 Interestingly, the high Tc of 98−107 K at 250 GPa has been also predicted in SiH4(H2)258 in which the direct interaction between H2 and SiH4 molecules at high pressure plays the major role in the superconductivity.58,59 The investigation of potential superconductivity in hydrogen-rich systems remains an interesting topic in high-pressure condensed matter physics. Here, we concentrate on GeH4(H2)2 to explore its structural, electronic, dynamical, and superconducting properties under pressure by first-principles calculations.

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COMPUTATIONAL DETAILS Referring to previous studies on similar hydrogen-rich systems, we have known that the situations of crystal structures containing one to four formula units (f.u.) per unit cell are frequent. Thus, we have searched the possible structures within the framework of about 200 space groups, and because of the difference of Wyckoff positions, we considered as many crystal configurations as possible for each space group. Candidate structures of GeH4(H2)2 contain one to four formula units per unit cell. Then, we calculated their total energies and enthalpies at the pressure points of 10, 50, 90, 130, 190, 250, 310, and 350 GPa. Selecting the relatively stable structures, we have extended

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RESULTS AND DISCUSSION Enthalpy−Pressure Phase Diagram. The phase stability of GeH4(H2)2 has been investigated systematically, based on first-principles calculations. Out of more than 3000 structures studied, nine new polymorphs of GeH4(H2)2 with low enthalpies are selected. Their enthalpy differences in the pressure range from 0 to 350 GPa are shown in Figure 1, and 5226



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Table 1. Lattice Parameters and Atomic Coordinates of GeH4(H2)2 for Several Important Phases of I4m ̅ 2, Pmn21, Pban, and P21/c at Different Pressures space group

pressure (GPa)

I4̅m2

8.5

Pmn21

Pban

P21/c

8.5

50

250

lattice parameters (Å/°) a = 4.490, c = 6.248 α=β=γ= 90

a = 4.484, b = 4.487, c = 6.261 α=β=γ= 90

a = 3.794, b = 3.254, c = 5.927 α=β=γ= 90 a = 2.914, b = 2.866, c = 5.932 α = γ = 90, β = 120.14

x

atom

y

z

Ge(2a)

0

0

0

H1(8i)

0.280

0.000

0.141

H2(4e) H3(4f) Ge(2a)

0 0 0

0 0.5 0.753

0.440 0.190 0.358

H1(4b)

0.280

0.754

0.217

H2(2a) H3(2a) H4(2a) H5(2a) H6(2a) H7(2a) Ge(2a)

0 0 0 0 0 0 0

0.473 0.034 0.708 0.780 0.234 0.294 0

0.500 0.499 0.803 0.911 0.163 0.051 0

H1(8m)

0.125

0.269

0.716

H2(8m) Ge(2d)

0.122 0.5

0.242 0

0.586 0.5

H1(4e)

0.971

0.125

0.847

H2(4e) H3(4e) H4(4e)

0.250 0.956 0.517

0.978 0.658 0.491

0.699 0.378 0.358

Figure 2. Energetically most favorable structures calculated for GeH4(H2)2 polymorphs at different pressures. The green and pink spheres represent Ge and H atoms, respectively.

whose enthalpy is only slightly higher than those of Pmn21 and I4m ̅ 2 phases, and the maximum enthalpy difference is 25 meV/ f.u. in the pressure range of 5−28 GPa. Upon compression, the P1̅ symmetry becomes more stable up to 48 GPa. This P1̅ structure of GeH4(H2)2 is similar to that of SiH4(H2)2 in the range of 50−90 GPa.58 For higher pressure in the range of 48− 150 GPa, GeH4(H2)2 exhibits the degenerated low enthalpy structures with Pban and Pbcn symmetries. The distinct layered character is observed in orthorhombic Pban and Pbcn structures as shown in Figure 2. After passing orthorhombic phase of Cmmm from 150 to 175 GPa, the GeH4(H2)2 is stabilized at the monoclinic P21/c phase up to 350 GPa. Low-Pressure Phases. We first focus on the low-pressure structures of GeH4(H2)2. The calculated equation of states (EOSs) of Pmn21, I4̅m2, and R3m phases are shown in Figure 3a. The same character of EOSs exists in I4̅m2 and Pmn21 phases, consistent with the nearly degenerated energies. The EOS of R3m structure is similar to those of I4m ̅ 2 and Pmn21 phases. However, the dynamical calculation for R3m shows an imaginary frequency at the Γ point in the BZ, which indicates the R3m phase is dynamically unstable. Comparing with that of SiH4(H2)2,55−57,59,60 the GeH4(H2)2 has a bigger volume at the same pressure. From structures shown in Figure 2, there are two GeH4(H2)2 formula units in each unit cell of Pmn21 and I4̅m2, while each unit cell of R3m phase consists of three formula units. These three low-pressure structures are constructed by weakly interacted GeH4 and H2 molecules, forming typical molecular solids. We notice that GeH4 molecules basically keep their original tetragonal packing with the orientating H2 molecules. These H2 molecules occupy the interstitial sites of Ge lattice. The calculated lattice parameters and inequivalent atomic coordinates of I4̅m2 and Pmn21 phases at 8.5 GPa are summarized in Table 1. To make the statement clearly, we define the dGe−H as the distance between Ge and H atoms in GeH4 or GeHx polyhedra, the dH−H as the H−H bond length in H2 units, the dH2−H2 as the nearest neighbor distance among H2 units, and the dGeH4−H2 or dGeHx−H2 as the nearest neighbor distance between GeH4 or GeHx polyhedra and H2 units, respectively. At 8.5 GPa, all dGe−H values in I4̅m2 and Pmn21 phases are 1.536 Å, which is slightly larger than 1.525 Å in germane. There are two kinds of H−H bond lengths of dH−H in both I4̅m2 and Pmn21 phases, 0.748 and 0.751 Å, which are formed by H2 and H3 atoms shown in Figure 3b. These values

Figure 1. Enthalpy difference versus pressure for competitive structures of GeH4(H2)2. The C2/c phase is taken as a reference point. The inset shows an enlargement of the low pressure region where the Pmn21 phase is taken as a reference point.

their atomic structures at favorable pressures are displayed in Figure 2. It is found that GeH4(H2)2 with Pmn21 and I4̅m2 symmetries are the most stable below 28 GPa. These are similar to the low-pressure structures of SiH4(H2)2.55,60 Especially, Pmn21 and I4m ̅ 2 structures are energetically nearly degenerated in a large pressure range, i.e., the enthalpy difference is less than 1 meV/f.u. in the range of 5−50 GPa. In addition, we have found another low-enthalpy structure with R3m symmetry 5227



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Figure 3. (a) Calculated equation of state of GeH4(H2)2 in I4̅m2, Pmn21, and R3m structures comparing with SiH4(H2)2.55,56 (b) The crystal structure of I4̅m2 phase are the green and pink spheres corresponding to Ge and H atoms, respectively. The dashed blue line denotes the distorted fcc lattice of Ge. (c) Raman spectra for the I4̅m2 and Pmn21 structures along with experimental data47 at 8.5 GPa, where main Raman peaks are presented.

are close to the dimer length of 0.74 Å. In both I4̅m2 and Pmn21 phases, the dGeH4−H2 is 2.227 Å, while the dH2−H2 is 2.362 and 2.328 Å, respectively. In trilateral R3m, each Ge site is equivalent, occupying the Wyckoff 3a sites, and part of the H atoms take the different Wyckoff 3a sites, while other H atoms occupy the Wyckoff 18c sites. As a result, the dH−H values are 0.747 and 0.749 Å, while the dGe−H values are 1.530 and 1.538 Å in R3m phase. For the low-pressure structures of GeH4(H2)2, X-ray diffraction patterns47 suggested an fcc structural model consisting of rotationally disordered GeH4 molecules located on the octahedral fcc sites (0,0,0) and rotationally disordered H2 molecules distributed between the other octahedral sites (0.5,0.5,0.5) as well as tetrahedral sites (0.25,0.25,0.25). Moreover, Raman and infrared spectra showed that the GeH4(H2)2 compound could be preserved to 27 GPa. Therefore, we examine the low-pressure stable structures in detail by constructing a larger cell denoted by dashed lines shown in Figure 3b for body centered tetragonal I4̅m2 structure at 8.5 GPa. We find that the lattice parameters a′ = b′ = 6.349 Å approximately are equal to c′ = 6.248 Å. In this case, the Ge lattice can be viewed as a slightly distorted fcc lattice. This idea also can be applied to the orthorhombic Pmn21 structure. Thus, the difference between I4̅m2 and Pmn21 is the orientations of H2 units. The same structure of Ge will be obtained by removing H2 molecules. As a result, it can naturally explain the experimental suggestion of an fcc structure.47 Comparing the predicted low-pressure phases of I4̅m2 and Pmn21 with experimental data, as mentioned above, the crystal lattice parameters are well consistent. Furthermore, we have calculated the Raman shifts of these two phases as shown in Figure 3c since Raman spectra are essential for identifying structures containing these structures. The Raman spectrum is clearly divided into three groups where the two low frequency groups are contributed by the vibrations of GeH4 molecules and the highest energy region is from H2 vibrations. Excellent

agreement between theory and experiment is found in these three frequency groups. Aiming at the order and disorder of H2 units in hydrogenrich systems, we also investigate the possibility of rotation of H2 molecules in GeH4(H2)2. Following the previous studies,55,56 in the case of I4̅m2 structure at pressures of 0, 10, and 30 GPa, we gradually rotate H2 units in (100) and (110) planes with keeping the center of H2 unchanged in calculations. The obtained energies relative to the initial state are viewed as the energy barrier during the rotation of H2, which are presented in Figure 4a,b, respectively. It is found that the maximum energy barriers of these H2 rotations increase from 0.72 meV/atom for rotation in (110) plane to 5.35 meV/atom for rotation in (100) plane with the pressure up to 30 GPa. These values are less than the maximum energy barrier of 12 meV/atom induced by the H2 rotation in hexagonal solid H2 at 35 GPa and correspondingly closed to those in SiH4(H2)2.55 This is due to the bigger distance between nearest neighbor H2 units in GeH4(H2)2 comparing to solid H2, as well as the bigger distance between H2 and GeH4. The obtained low energy barriers imply a free rotation of H2 in GeH4(H2)2 and an orientational disorder. This result is consistent with the experimental suggestion of disorder.47 In spite of van der Waals interactions in low-pressure structures of GeH4(H2)2, the reliability of density functional theory has been established to deal with the related systems.55 Furthermore, the molecular dynamics calculation also established the disorder of H2 in similar systems, such as SiH4(H2)2.56,57 Seen from Figure 4a,b, the energy barriers increase with the pressure, which implies an ordering of H2 units in GeH4(H2)2 at higher pressures. Namely, these H2 units have difficulty in rotating at higher pressures. Additionally, the ordering of H2 was observed in Xe(H2)753 and Ar(H2)275 at high pressures. For Xe(H2)7, Ar(H2)2, SiH4(H2)2, GeH4(H2)2, and solid H2, the ordering or disordering strongly depends on the distance between X (X represents SiH4, GeH4, Xe, and Ar.) and H2 as well as the distance between two nearest 5228



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dGeH4−H2 shortens to 1.751 Å at 40 GPa, implying the reduction of van der Waals interactions. However, the P1̅ phase, together with I4̅m2, Pmn21, and R3m phases, exhibits the insulating behavior at low pressures. Upon compression, the insulator-tometal transition occurs in GeH4(H2)2 at 48 GPa, accompanied by the P1̅ phase transforming to Pban phase. Figure 5a presents the pressure dependence of the band gap in GeH4(H2)2. It is found that the band gap rapidly decreases from 2.06 eV to 0 as the pressure increases from 40 to 48 GPa and that GeH4(H2)2 shows the metallic behavior above the pressure range of 48− 350 GPa, though two structural phase transitions occur by further compression. As mentioned above, the GGA underestimates the band gap of GeH4(H2)2. Namely, GGA also underestimates the metallization pressure. It has proved that the metallization pressure predicted by GWA is larger than that by GGA.73,74 For example, the metallization pressures predicted by GWA is 164 GPa, which is larger than 145 GPa obtained by GGA.73 Therefore, the metallization pressure of GeH4(H2)2 obtained from GWA should be less than that from GGA. Considering the different effects between GGA and GWA; however, this metallization pressure of GeH4(H2)2 is much less than 200 GPa,59 145 GPa,73 120 GPa,56 or 92 GPa58 of SiH4(H2)2. As shown in Figure 5b,d, the band gap of interesting structures first decreases with the dH−H slightly contracting from 0.749 to 0.745 Å and then suddenly disappears at a relatively large bond length of 0.775 Å (at 48 GPa). After transferring to metal, on the contrary, the dH−H increases except for an occasional decrease in the pressure range of 150−175 GPa. Obviously, the band gap decreases with the dGeHx−H2 shortening. Seen from the dH2−H2 values, the band gap decreases with the dH2−H2 shortening and closes at 1.380 Å, as shown in Figure 5b. At the same time, Figure 5c,d illustratively exhibits the evolution of dH2−H2, dH−H, dGeHx−H2, and dGe−H with pressure.

Figure 4. (a,b) Energy barriers during the orientating rotation of H2 in (100) and (110) planes at 0, 10, and 30 GPa, respectively.

neighbor H2 units. If this distance is smaller than that in solid H2, the ordering becomes easy and vice versa. Pressure-Induced Metallization. The P1̅ phase, another low-pressure structure existing in the range of 28−48 GPa, is also a molecular solid as shown in Figure 2. The dH−H values are 0.745 and 0.749 Å at 40 GPa, respectively, which are slightly smaller than those of the former three low-pressure structures of GeH4(H2)2. Surprisingly, the dGe−H is slightly compressed to 1.536−1.522 Å for P1̅ phase with increasing pressure, and the

Figure 5. (a) Calculated magnitudes of band gap for GeH4(H2)2 as a function of pressure. (b) Band gap of GeH4(H2)2 dependence on the shortest H−H bond length in H2 units (dH−H), and the inset shows the dependence of band gap on the nearest neighbor distance among H2 units (dH2−H2) and the nearest neighbor distance between GeHx polyhedra and H2 units (dGeHx−H2). (c) The dH2−H2, the nearest neighbor distance between Ge and H atoms (dGe−H), and dGeHx−H2 vary with the pressure. (d) The dH−H changes with the pressure. The data was chosen from I4̅m2 below 28 GPa, P1̅ in 30−48 GPa, Pban from 50 to 150 GPa, Cmmm in 150 < P < 175 GPa, and P21/c above 175 GPa, respectively. 5229



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frequency of ∼129 cm−1 at 50 GPa, which indicates a dynamic instability in Pban phase. Hence, the possibly decomposed enthalpies in Ge + 4H2 and GeH4 + 2H2 paths were examined to check the phase stability of GeH4(H2)2 under pressure. In the decomposed enthalpy calculations, structures of P63m at 0− 105 GPa, C2/c at 105−270 GPa, and Cmca1̅2 above 270 GPa for H2 were used.76 For Ge, structures of Fd̅3m at 0−10 GPa, I41/amd at 10−67 GPa, Imma at 67−90 GPa, P6/mmm at 90− 110, Cmca at 110−170 GPa, and P63/mmc above 170 GPa were adopted.77 For GeH4, structures of P21/c at 0−15 GPa, Cmmm at 15−40 GPa, P21/m at 40−150 GPa, and C2/c above 150 GPa were chosen,27 though the structure of GeH4 above 200 GPa was still dubious. The enthalpy pathways of GeH4(H2)2 decomposing into Ge + 4H2 and GeH4 + 2H2 are also shown in Figure 1. Interestingly, we find that all the structures below 220 GPa are metastable relative to the decomposition and only the P21/c structure becomes stable above ∼220 GPa. The GeH4(H2)2 easily decomposes into Ge plus four H2 below ∼155 GPa, while it possibly decomposes into GeH4 plus two H2 in the pressure range of 155−220 GPa. For comparison, this result is different from experiment47 where the GeH4(H2)2 compound can be preserved up to 27 GPa and then decomposes. This phenomenon of decomposing at low pressures while stabling at high pressures is also existent in other theoretical studies of hydrogen-rich systems, such as GeH4,25 SiH4,12 and SiH4(H2)2.58 It is possible that the experiments were performed at finite temperature but the firstprinciples calculations essentially describe the ground states at 0 K. At relatively low temperatures, a large kinetic barrier might exist and prevent GeH4(H2)2 from decomposition. Some studies on hydrogen containing systems have revealed fairly large activation energies on decomposition, e.g., 79.3−102.2 kJ/ mol for Al−H,78 71−74 KJ/mol for Mg−H,79 and 78 KJ/mol for Pd−Ni−H.80 An accurate evaluation of the decomposition kinetic barrier is needed to give a precise conclusion but remains a major challenge and beyond the scope of this study. Additionally, quantum effects related to hydrogen atoms have been analyzed in the selected cases. Particularly, the hydrogen zero-point (ZP) energy is expected to be large enough to significantly revise the structural stability as in the cases of solid hydrogen and hydrides. We therefore have estimated the ZP energies of GeH4(H2)2, GeH4, and H2 at 250 GPa using the quasi-harmonic approximation.81 It is found that ZP effects do not change the topology of the phase diagram but has the enthalpy difference between GeH 4 (H 2 ) 2 and GeH4+2H2 enlarged. The ZP energies are 0.698, 0.375, and 0.179 eV/f.u. for GeH4(H2)2, GeH4, and H2 at 250 GPa, respectively. Namely, the ZP energy of GeH4 + 2H2 is 0.733 eV/f.u. After adding ZP energy corrections based on initial energy values, the enthalpy difference between GeH4(H2)2 and GeH4 + 2H2 becomes 0.068 eV/f.u., which is larger than 0.033 eV/f.u., and the enthalpy of GeH4(H2)2 is still lower than that of GeH4 + 2H2. It means that the GeH4(H2)2 is more stable when considering the ZP effects. High-Pressure Stable Phase. Structural and Electronic Characteristics. For the high-pressure stable P21/c phase of GeH4(H2)2, we now discuss its crystal structural, electronic, dynamical, and superconducting properties in detail. Geometrical structure and crystal lattice parameters as well as atomic coordinates of P21/c phase at 250 GPa are shown in Figure 2 and Table 1, respectively. From the geometrical structure, the P21/c consists of two GeH4(H2)2 formula units. All H atoms form the H2 units, and there are two kinds of H2

In the low-pressure region, the dH2−H2 rapidly decreases. The decrease becomes slow after transforming into metal, but an abnormality is observed near 160 GPa, which is correlated to the complex structure in that region. With regard to the variations of dGe−H shown in Figure 5c, the rapid increase near 50 GPa results from the transition from insulator to metal and the change of H coordination number around Ge atom. At low pressures, the Ge−H is in the approximate tetrahedral form with the dGe−H value closed to that in germane. Around the insulator-to-metal transition point, an obvious layered structure is formed with the Ge and H atoms separating. At high pressure, Ge−H exists in the more coordination number, such as GeH8 polyhedra shown in Figure 2. The dGeHx−H2 mainly decreases with the increase of pressure. The variation of dH−H is also complex as shown in Figure 5d. Below 48 GPa, the dH−H slightly decreases. In contrast, it increases above 48 GPa, but there is jumping and an abnormal decrease at around 180 GPa. At 250 GPa, the dH−H stretches to 0.839 and 0.873 Å. We also choose several pressure points and investigate the interatomic bonding and interaction. Figure 6 shows the

Figure 6. Calculated total charger densities for I4m ̅ 2 phase at 8.5 GPa (a), Pban phase at 50 GPa (b), and P21/c phase at 250 GPa (c) on the plane containing Ge, Ge−H, and H2 atoms or bonds. The color schemes from blue to red represents the density from low to high.

calculated total charge densities on the plane containing the Ge, Ge−H, and H2 units at pressures of 8.5, 50, and 250 GPa. It is visibly found that the intramolecular interactions are stronger than the intermolecular ones for H2 and GeH4 units at 8.5 GPa as shown in Figure 6a, which results in a typical molecular solid in GeH4(H2)2. Upon compression to 50 GPa as shown in Figure 6b, the intermolecular interaction strengthens for interlayered H2 units with forming the weak covalent bond, and in Ge atomic layers, the clear metallic bonding is formed. GeH4(H2)2 exhibits the metallic characteristic. At 250 GPa as shown in Figure 6c, although the intramolecular H−H interaction in H2 units slightly reduces, the covalent bonding interaction dominates in the interior of all H2 units as well as between Ge and H2 units. These interactions reflected by the charge densities in Figure 6 are consistent with the analysis above of interatomic bonds and distances. Thus, the stability of GeH4(H2)2 mainly comes from the strengthening of the intermolecular interaction with the increase of pressure. Decompositions in GeH4(H2)2. To seek for new superconducting phases for GeH 4 (H 2 ) 2 with the moderate metallization pressure, we have checked the Pban phase. However, the dynamics calculation shows the big negative 5230



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−25 to 15 eV. In the energy region from −4 to 4 eV, the electronic states are collectively contributed by Ge, H1(H2), and H3(H4) atoms, but the contribution of Ge is obviously stronger than those of H1(H2) and H3(H4) atoms. The P21/c symmetric GeH4(H2)2 at 250 GPa exhibits the rich and multiple Fermi surface feature due to the complex electronic band structure near the EF. As shown in Figure 7, band 1 forms the hole-like character around the Y k-point and along the D−E direction in the BZ, and bands 3 and 4 form the electron-like character in the center of the A k-point in the BZ, while band 2 results in a multiple Fermi surface sheet containing both electron-like and hole-like characters. These rich Fermi surface features are favorable to the strong electron− phonon interaction in a superconductor. Furthermore, the characterized band structure from more fine electronic structures indicates that flat bands around the Y k-point mainly result from the mixing between Ge px and H4 s states, while the flat bands along the B−D direction in the BZ are induced by the mixing between Ge pz and H3 s as well as H1 s states. This means that the possible superconductivity is not only related to all H2 units but also the interaction between Ge and H atoms. Phonon Spectrum and Superconductivity. As mentioned above in the calculated barriers shown in Figure 4, it is proposed that the H2 units have difficulty in rotating and begin ordering at higher pressures. Therefore, when analyzing the vibrational and superconducting properties of P21/c phase, the possibility of rotation and disorder of H2 units in GeH4(H2)2 has been ignored. Figure 8 shows these calculated results. The

units in this system. The dH−H values in H2 units are 0.839 and 0.873 Å at 250 GPa, respectively, which are somewhat larger than the dimer length of 0.74 Å. The distance of H−H bond in H2 units becomes bigger, which results in the strengthening of electronic delocalization. It is related to metallization. The H2 units with the H−H bond length of 0.873 Å are formed by H1 and H2 inequivalent atoms shown in Table 1, while the other H2 units are formed by H3 and H4 inequivalent atoms. GeH4 tetrahedrons are broken. All Ge atoms are in (020) plane with the nearest neighbor Ge−Ge distance of 2.836 Å. The H2 units formed by H1 and H2 atoms are parallel to the (010) plane and are localized among Ge atoms. The H2 units formed by H3 and H4 atoms result in the ordering chain-like structure along the b direction between two Ge atomic planes. Interestingly, H1 and H2 atoms not only form the H2 units but also form the GeH8 octahedra with Ge atoms, namely, one Ge atom interacts with the other Ge atom by means of H2 units formed by H1 and H2. At 250 GPa, all the Ge−H bond lengths in GeH8 octahedra are almost equal to 1.70 Å. Figure 7 shows the calculated electronic band structure along high-symmetric k-point directions in the BZ and density of

Figure 7. Calculated electronic structures for P21/c GeH4(H2)2 phase at 250 GPa. (a) Electronic band structure and density of states (DOS) where the EF is set as zero. Four bands crossing the EF are marked by 1, 2, 3, and 4, respectively. (b) The BZ and the whole Fermi surface are shown, and the Fermi surface features corresponding to those bands crossing the EF are distinguished.

states (DOS) projected on atoms as well as the threedimensional view of the Fermi surface characteristics at 250 GPa. The electronic band structure reveals the metallic feature with four bands (respectively marked by 1, 2, 3, and 4 as shown in Figure 7a) crossing the Fermi level (EF). Note that there are platforms in these four bands near the EF around the Y k-point and along the B−D direction in the BZ. These flat bands near the EF result in the large DOS of 0.91 states/eV/cell, which will be propitious to forming enough Cooper electron pairs for a high-temperature superconductor. The projected DOS on atoms shows the strong hybridization among H2 units as well as between Ge atoms and H2 units in the energy range of from

Figure 8. (a) Calculated phonon band structure, (b) phonon density of states (PhDOS) projected on each atom, and (c) the Eliashberg phonon function α2F(ω) and the electron−phonon integral λ(ω) for P21/c GeH4(H2)2 phase at 250 GPa.

phonon band structure shown in Figure 8a confirms the stability of P21/c GeH4(H2)2 at 250 GPa due to the absence of the imaginary frequency mode. By combining the phonon density of states (PhDOS) projected on atoms shown in Figure 8b, we find that the low-frequency vibrations below 290 cm−1 mainly come from the vibrations of Ge as well as the weak coupling between Ge and all H atoms. The strong coupling 5231



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reflected by the coupling between Ge p states and all H2 units shown in Figures 7 and 8. In addition, comparing the crystal structures of GeH4(H2)2 with other hydrogen-rich systems, there are more H2 units in GeH4(H2)2 than SiH4, GeH4, and SnH4. So, the higher Tc is obtained in GeH4(H2)2. The interaction between SiH8 and H2 units in SiH4(H2)2 is stronger than that in GeH4(H2)2, though more H2 units are formed in the latter. Hence, similar superconducting transition temperatures are obtained in these two hydrogen-rich compounds.

between H1 (H2) and H3 (H4) covers the frequency region of 290−2040 cm−1 and the high-frequency above 2240 cm−1. Three separate regions of phonon band/states can be recognized due to the existence of a gap. Similar to C2/c symmetric GeH4 at 220 GPa,25 there is a pseudogap in the lowfrequency region, while there is a gap of 200 cm−1 in the highfrequency region, which probably comes from the layered structure in P21/c GeH4(H2)2, just like in the case of C2/c GeH4. No visible frequency gap is found in Ccca SiH4(H2)2 at 250 GPa.58 Noticeably, the high-frequency gap in P21/c GeH4(H2)2 is far less than those of Cmmm GeH4 (∼2500 cm−1 at 20 GPa)26,27 and P6/mmm SnH4 (∼800 cm−1 at 120 GPa).28,30 Examining the dH−H values, we find that they are 0.839 and 0.873 Å in P21/c GeH4(H2)2 phase at 250 GPa, while it is 0.843 Å in Ccca SiH4(H2)2 at 200 GPa, and further stretching at 250 GPa. This bond length is 0.746 Å in Cmmm GeH4 at 20 GPa and is 0.82−0.79 Å in Ama2 SnH4 at 96−120 GPa. As a result, the magnitude of the frequency gap is mainly caused by the variation of dH−H, namely, this bond length is closer to the dimer length of 0.74 Å; the coupling among atoms is weaker, and then, the gap is larger. The Eliashberg spectral function α2F(ω) and the electron− phonon coupling integral λ(ω) of the P21/c phase as a function of frequency at 250 GPa were calculated to explore the possible superconductivity of GeH4(H2)2. The results are shown in Figure 8c. The total λ is ∼1.43 at 250 GPa, which means that the electron−phonon interaction is very strong. From the phonon band structure and PhDOS shown in Figure 8, we find that the vibrational modes from two kinds of H2 units in frequency ranges of 290−2040 and 2240−2800 cm−1 mostly contribute (75%) to the total λ value. The low frequencies below 290 cm−1, which are from the vibrations of Ge atoms by coupling with all H atoms, contribute the remaining 25% of the total λ(ω) parameter. On the basis of the obtained α2F(ω) and λ(ω), we now can analyze the superconductivity using the modified McMillan equation by Allen and Dynes82 Tc =

⎞ ⎛ 1.04(1 + λ) exp⎜ − ⎟ * 1.2 ⎝ λ − μ (1 + 0.62λ) ⎠

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CONCLUSIONS

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AUTHOR INFORMATION

Starting from the first-principles calculations, we have studied structure, phase transition, bonding interaction, electronic structures, dynamic stability, and superconductivity of hydrogen-rich GeH4(H2)2 under the pressure condition. In lowpressure less than 28 GPa, I4m ̅ 2 and Pmn21 symmetries, two degenerated phases, are the most favorable. They both are viewed as the distorted fcc structure as mentioned by experiment. On compression up to 48 GPa, the P1̅ symmetry is more preferred. At low pressures, although the bond lengths of Ge−H and H−H in H2 units slightly change, GeH4 and H2 units keep the characteristics of germane and hydrogen molecules. The H2 units are disordered or rotated around GeH4 at low pressures, but they become ordered with the increase of pressure. In the pressure range of 48−150 GPa, GeH4 and H2 possibly exists in the form of Pban or Pbcn phase, which is also degenerated. When increasing pressure, the system transfers to Cmmm symmetry in the small range of 150−175 GPa. Then, GeH4(H2)2 is stabilized at the P21/c phase up to 350 GPa. The P21/c structure consists of H2 units and Ge atoms, simultaneously forming the GeH8 polyhedra between Ge and part of the H atoms. The H2 intermolecular and intramolecular distances decrease and increase with the increase of pressure, respectively. The Ge−H bond goes through the tetrahedral germane form, the separating of Ge and H atoms, and the polycoordinated GeHx (x > 4) form. With the increase of pressure, the van der Waals interactions of H2− H2 and GeH4−H2 weaken, and GeH4(H2)2 exhibits a metallic feature with the forming of a Ge−Ge metallic bond above 48 GPa. In the end, covalent bonding dominates the compound, which results in the stability of GeH4(H2)2 at high pressures. However, the decomposition enthalpies indicate that the GeH4(H2)2 is decomposed into Ge and H2 below 155 GPa and GeH4 and H2 in the pressure range of 155−220 GPa. GeH4(H2)2 is only stable above 220 GPa in P21/c phase. The P21/c symmetric GeH4(H2)2 is metallic and dynamically stable. A strong electron−phonon interaction and a high superconducting transition temperature are obtained in this system. At 250 GPa, the electron−phonon coupling constant in GeH4(H2)2 reaches ∼1.43, which leads to the high transition temperature of 76−90 K. The superconductivity mainly results from the vibration of more H2 units, though the vibrations of Ge at low frequencies also contributes to it by coupling with H2 units.

ω log

(1)

where ωlog is the logarithmic average of phonon frequencies, and μ* is the Coulomb pseudopotential representing Coulombic repulsion. With the typical choice of μ* as 0.1− 0.15, we obtain the high superconducting transition temperature Tc in the range of 76−90 K of GeH4(H2)2 at 250 GPa. This superconducting transition temperature of GeH4(H2)2 is somewhat less than 98−107 K of SiH4(H2)2 at 250 GPa.58 However, the result is obviously higher than those of previous reported hydrides15,17,25−27,31 such as SiH4, GeH4, and SnH4. Noticeably, the calculated superconducting transition temperature highly depends on the choice of the Coulomb pseudopotential μ*, so we only present an usual range. Moreover, whether the conventional BCS theory is fitting for hydrogen-rich systems, which needs more research work to prove, a strong electron−phonon interaction of λ ≈ 1.43 is obtained. To further analyze the superconductivity in P2 1 /c GeH4(H2)2, we review the P21/c structure and phonon density of states as shown in Figures 2 and 8, respectively. We can conclude that the high-temperature superconductivity mainly results from the vibrations of H2 units. Besides, at the lowfrequency region, the vibrations of Ge atoms in the grid formed by H2 units also contribute to superconductivity, which is

Corresponding Author

*Phone: +86-755-86392132. Fax: +86-755-86392299. E-mail: [email protected]. 5232



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ACKNOWLEDGMENTS The work was supported by the RGC Grant (No. CUHK 402108), NSFC Grants (Nos. 10874046, 10947169, and 11047013) and National Basic Research Program of China (973 Program) under Grant Nos. 2012CB933700 and 2011CB922200. Work done in the U.S. was supported by the Department of Energy under Grants No. DE-SC0001057 (EFree) and No. DEFC03-03NA00144 (CDAC). The calculations were performed in HPC Lab, SIAT, CAS, China.

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