Ab initio study of germanium-hydride compounds under high pressure

Abstract

Motivated by the potential high-temperature superconductivity in hydrogen-rich materials and phase transitions, germanium-hydride compounds under high pressure were studied by a genetic algorithm. Enthalpy calculations suggest that the Ge and H will form Ge₃H, Ge₂H, GeH₃, and GeH₄ at about 32, 120, 280, and 280 GPa, respectively. These four germanium-hydride compounds are all stable up to at least 300 GPa. For Ge₃H, the most stable structure is P3̄-Ge₃H at 32–220 GPa and P6₃/m-Ge₃H at 220–300 GPa. All the germanium-hydride compounds are metallic phases as demonstrated by the band structure and density of states.

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I. Introduction

In the past few decades, scientists have been trying to find the best way to seek and design high-temperature superconducting materials.^1–3 And in the high-pressure field, one of the most significant targets is to discover metallic hydrogen for high-temperature superconductor and the other one is to achieve production, transition and application of hydrogen. Theoretical works have reported that hydrogen will metalize at about 440 GPa. However, experimental metallization is not observed up to 342 GPa,⁴ but will lower a T_c because of chemical pre-compression.⁵ Recently, it was suggested that dense hydrogen-dominated metallic alloys, in strongly compressed group IV hydrides, can be potential candidate materials for a high-temperature superconductor. Disilane has been studied by Jin et al.⁶ and they explored the crystal structures of disilane in a wide pressure range from 50 to 400 GPa, and three favored structures were P1̄, Pm3̄m, and C2/c. Remarkably, the large T_c of 80 K at 200 GPa for P1̄ and 139 K at 275 GPa for Pm3̄m are predicted by quantitative calculations. Germane has been forecast by Gao et al.,⁷ and their calculated results suggested a remarkably wide decomposition pressure range of 0–196 GPa, above which a C2/c structure is stable. Then GeH₄ (ref. 8) and GeH₄(H₂)₂ (ref. 9) were studied by Chao Zhang and Guohua Zhong et al. GeH₃ has been predicted to be a Cccm structure by Kazutaka Abe et al.,¹⁰ and the calculation of enthalpy indicates that Cccm GeH₃ can be formed at about 200 GPa.

In order to search out the real conditions of Ge₃H, Ge₂H and GeH₃, we extensively explored the structures of germanium-hydride compounds under high pressure using an ab initio evolutionary algorithm^11,12 for crystal structure prediction. We propose the compound Ge₃H has more potential in the pressure range of 40–300 GPa. Enthalpy calculations suggest that the P3̄ structure is the most favorite one at 40–220 GPa and the P6₃/m structure is the most favorite one at 220–300 GPa. Calculations of DOS indicate that Pnma Ge₂H, P3̄ Ge₃H, P6₃/m Ge₃H and Cccm GeH₃ are all metallic. For Cccm GeH₃, as an underlying candidate for a high-temperature superconductor, the application of the Allen–Dynes modified McMillan equation¹³ reveals a high T_c of 80.3 K at 300 GPa for this structure. Our calculations have also revealed that there will be a P3̄ Ge₃H under high pressure which has been considered not to form any compounds of Ge and H below about 32 GPa. We predict that Ge and H will form GeH₃ above 280 GPa. At the same time, it is clear that another metallic germanium-hydride structure, C2/c GeH₄,⁷ will not form below 280 GPa.

II. Computational method

An ab initio evolutionary algorithm^10–12 designed to search for the structure possessing the lowest free energy at given pressure conditions was employed. The most significant feature of this method is the capacity to predict a stable structure with only the knowledge of the chemical composition.⁵ The details of the search algorithm and its initial several applications have been described elsewhere. The underlying ab initio structure relaxations were performed using density functional theory within the Perdew–Burke–Ernzerhof (PBE) parameterization of the generalized gradient approximation (GGA), as performed in the Vienna ab initio simulation package VASP code to calculate projected density of states.^14,15 The PAW¹⁶ potential with the valence states 1s¹ for H and 4s²4p² for Ge¹⁷ has been employed with plane waves up to a cutoff energy of 720 eV, and the density of states has been calculated with a 8 × 8 × 9 k-point mesh created by the Monkhorst–Pack (MP)¹⁸ method. Phonon dispersion and electron–phonon calculations were performed with the density functional perturbation theory^19,20 using the program QUANTUM ESPRESSO.²¹ Convergence tests gave the choice of a kinetic energy cutoff of 72 Ry and 8 × 8 × 9 Monkhorst–Pack grids of k-point sampling for the electronic Brillouin Zone integration. We calculated T_c using the Allen–Dynes equation. The q-point mesh in the first BZ of 4 × 4 × 4 for a Cccm structure is used in the interpolation of the force constants for the phonon dispersion curve calculations. A denser k-point mesh, 12 × 12 × 12, for Cccm structure was adopted to ensure k-point sampling convergence with a Gaussians width of 0.03 Ry, which approximates the zero width limits in the calculation of the EPC parameter λ. The hybrid functional calculations are implemented in the CASTEP code, together with norm-conserving pseudopotentials, and a cutoff energy of 600 eV.^22,23

III. Results and discussion

We performed variable-cell structure prediction simulations using the above evolutionary methodology for one, two, three, and four molecules in contained the simulation cell at 10, 30, 50, 100, 150, 200, 250 and 300 GPa, respectively. At 50 and 100 GPa, our simulations predicted a P3̄ structure as depicted in Fig. 1(a). The Ge and H atoms are depicted in violet and pink colors, respectively. Each H atom connects with three Ge atoms. Both kinds of atoms form the layered structure. Fig. 2(b) shows the enthalpy curves for the P3̄ Ge₃H structure, and the decomposition of Cmca Ge and C2/c H₂ (ref. 24–26) with respect to our predicted P3̄ structure. Obviously, the currently proposed structures are much superior in enthalpy than the other structures. At 200 and 250 GPa, our simulations predicted a P6₃/m structure as depicted in Fig. 1(b), and at 100 and 150 GPa a Pnma structure as depicted in Fig. 1(c). It is clear that Cccm GeH₃,¹⁰ as depicted in Fig. 1(d), will not be easily formed until 280 GPa and Ge and H will form P3̄ Ge₃H at 32 GPa. Enthalpy calculations reveal that the Cccm structure is stable at 280 GPa up to at least 300 GPa, while P3̄ Ge₃H and Pnma Ge₂H are the most stable two phases from 120 GPa to 220 GPa. From 220 GPa to 280 GPa, Pnma Ge₂H and P6₃/m Ge₃H will coexist. No imaginary phonon frequencies were found in the Brillouin Zones which confirms the P3̄ Ge₃H structural stability of the new stable structure from 40 GPa to 220 GPa, P6₃/m Ge₃H from 220 GPa to 300 GPa and Pnma Ge₂H from 50 GPa to 300 GPa. Due to the chemical precompression,⁵ the GeH₃ may be a good superconductor under high pressure. To explore the superconductivity of the Cccm GeH₃, the application of the Allen–Dynes modified McMillan equation¹³ reveals a high superconducting temperature of 80.3 K for the Cccm phase at 300 GPa and 100.8 K at 220 GPa. At the stability field of P3̄, Pnma and P6₃/m, the electronic properties, lattice dynamics, and electron–phonon coupling of the P3̄, Pnma and P6₃/m structures are explored. In accordance with our expectation, the superconduction transition temperatures of these three phases are very low as a result of the low content of hydrogen. This result maybe proves that, to a great extent, the superconduction transition temperature depends on the content of hydrogen in a metal hydride (Table 1).

Fig. 1 (a) Crystal structure of Ge₃H with the P3̄ space group at 40 GPa. (b) Crystal structure of Ge₃H with the P6₃/m space group at 220 GPa. (c) Crystal structure of Ge₂H with the Pnma space group at 200 GPa. (d) Crystal structure of GeH₃ with the Cccm space group at 300 GPa.

Fig. 2 Convex hull diagram for the Ge–H system. At pressures of (a) 40 GPa, (c) 60 GPa, (d) 120 GPa, (e) and (f) 220 GPa, (g) 280 GPa and (h) 300 GPa respectively. Enthalpy curves of P3̄ Ge₃H (b).

Table 1 The parameters of P3̄ structures correspond to 40 GPa, Pnma structures correspond to 200 GPa, P6₃/m structures correspond to 280 GPa

Pressure (GPa)	Space group	Lattice parameter (Å)	Atomic coordinates
40	P3̄	a = b = 4.97724
		c = 4.114922	Ge1	0.65731	0.70024	0.24985
		α = β = 90°	H1	0.66667	0.33333	0.25031
		γ = 120°
200	Pnma	a = 4.559771
		b = 3.534107	Ge1	0.12917	0.75000	−0.10391
		c = 5.306068	Ge2	1.04057	0.75000	−0.65622
		α = β = γ= 90°	H	0.25837	0.75000	−0.40183
220	P63/m	a = b = 4.403848
		c = 3.716815	Ge	−0.04740	−0.70651	0.25000
		α = β = 90°	H	0.33333	−0.33333	0.25000
		γ = 120°

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To further confirm the dynamical stability of Ge₃H and Ge₂H, we calculated the phonon dispersion curves, as shown in Fig. 3. That no imaginary phonon frequencies were found in the Brillouin Zones confirms the structural stability of the new structures. The calculated energy band structures and density of states of Ge₃H and Ge₂H are shown in Fig. 4 and Fig. 5, respectively. The overlap between the conduction and the valence bands suggests that Ge₃H and Ge₂H are metallic. The electronic band structure and density of electronic states (DOS) of Cccm GeH₃ are shown in Fig. 6, which shows a significant overlap between the orbitals of s and p electrons. The band structure reveals a metallic character with large dispersion bands crossing the Fermi level (E_F) and a flat band around E_F. The simultaneous occurrence of flat and steep bands near the Fermi level has been suggested as a favorable condition for strengthening the electron and phonon coupling, which is essential to superconducting behavior.⁷ Due to an underestimation of the band gaps of the GGA functional, we also calculated the band structures of germanium-hydride compounds with the hybrid functional HSE06 method. The band structures calculated by the hybrid functional HSE06 method are the same as the GGA results, suggesting that the structures at the pressures are all metallic phases.

Fig. 3 The phonon dispersion curves for the P3̄ Ge₃H structure configuration at pressures of 40 GPa (a), 220 GPa (b), Pnma Ge₂H structure configuration at pressures of 50 GPa (c), 300 GPa (d) and the P6₃/m Ge₃H structure configuration at pressures of 220 GPa (e), 300 GPa (f).

Fig. 4 The electronic band structure of P3̄ Ge₃H at 40 GPa (a), 220 GPa (b), Pnma Ge₂H at 150 GPa (c), 280 GPa (d) and P6₃/m Ge₃H at 220 GPa (e), 280 GPa (f).

Fig. 5 The density of electronic states (DOS) of P3̄ Ge₃H at 40 GPa (a), 200 GPa (b), Pnma Ge₂H at 180 GPa (c), 240 GPa (d) and P6₃/m Ge₃H at 220 GPa (e), 280 GPa (f).

Fig. 6 The electronic band structure and density of electronic states (DOS) of Cccm GeH₃.

According to the McMillan formula,¹³ T_c is governed by three parameters: coupling-weighted phonon momentum, electron–phonon coupling, and conduction electrons near the E_F. Conduction electrons near the E_F are ruled out as the main contributor for the superconductivity behavior, since there is only a low density of Ge s electrons and an even lower one for H s electrons. But the shape of the density of the H s electrons is much like that of the Ge s electrons, which helps to enhance the electron–phonon coupling. The change in coupling-weighted phonon momentum under pressure is seen to be consistent with that of T_c. This leaves electron–phonon coupling as the dominant factor that affects T_c. We thus explain the likeness between the respective behaviors of T_c and the hybridization function under pressure to be because of an improvement in enhancing electron–phonon coupling by the match of the energy level from the Ge p electrons with the peak of the H p electrons. The peak of the Ge p electrons is found to be above E_F, and is found close to the E_F and can therefore contribute strongly to the electron–phonon coupling. The calculated electronic band structure and projected density of states (DOS) for Cccm GeH₃ at 280 GPa presented in Fig. 6 reveal that this structure is metallic. However, the less dispersed valence and conduction bands near the Fermi level represent a rather large electronic DOS at the Fermi level, which might favor a superconducting behavior. The calculated valence bandwidth is consistent with earlier predictions and recent theoretical results for dense hydrogen alloys.⁵ The strong Ge–H hybridization can be gained from the significant overlap of Ge-DOS and H-DOS. Phonon calculations performed at 280 and 300 GPa have proved the dynamical stability of the Cccm structure by evidence of the absence of any imaginary frequency modes in the BZ. To explore the superconductivity of this phase we suggested that the EPC parameter λ, the logarithmic average phonon frequency (ω_log), and the Eliashberg phonon spectral function α²F(ω)²⁶ be investigated at high pressures. The resulting λ for this phase at 220 and 300 GPa are 1.65 and 1.36, respectively, indicating that the EPC is fairly strong. The theoretical spectral function α²F(ω)²⁶ and the integrated λ(ω) as a function of frequency at selected pressures are shown in Fig. 7. The superconducting critical temperature can be estimated from the Allen–Dynes modified McMillan equation, which has been found to be highly accurate for many materials with λ < 1.5. The ω_log is calculated directly from the phonon spectrum. The Coulomb pseudopotential μ is often taken as 0.1 for most metals; an appropriate one proposed by Ashcroft is 0.12 for hydrogen-dominant metallic alloys and has been adopted in the works of GeH₄ and SnH₄. These current studies inevitably stimulate future high-pressure experiments on structural and conductivity measurements.

Fig. 7 Eliashberg phonon spectral function α²F(ω) and the electron–phonon integral λ(ω) (a), and density of phonon state PHDOS (b) for phase Cccm calculated at 300 GPa.

Extra phonon calculations show the stability range to be between 280 and at least 300 GPa.¹⁰ Obviously, for a low frequency area, Ge contributes a lot to phonon DOS; by contrast, for a high frequency all the contributions come from the electrons of the H atoms.^27,28 In general, the integrable function depends on the high frequency, which is contributed by the electrons of H atoms.

IV. Conclusions

In summary, the structures of germanium-hydride are explored with the ab initio evolutionary method. The P3̄ Ge₃H structure is the most stable one in the pressure range from 40 GPa to at least 220 GPa, the Pnma Ge₂H structure will coexist with P3̄ in a pressure range from 120 GPa to at least 220 GPa and coexist with P6₃/m Ge₃H from 220 GPa to 300 GPa. And the ab initio calculation of energy shows that Ge₃H and Ge₂H will be formed more easily than GeH₃ below 280 GPa. The calculated band structures of these four phases reveal a metallic character, specially, for Cccm GeH₃, with large dispersion bands crossing the Fermi level (E_F), which is necessary for superconductive behavior. Electron–phonon coupling calculations show that this phase is superconducting with a high T_c of 80.3 K at 300 GPa. The current study will inevitably stimulate future high-pressure experiments on structural and conductivity measurements.

Acknowledgements

We are thankful for financial support from the National Basic Research Program of China (no. 2011CB808200), Program for Changjiang Scholars and Innovative Research Team in the University (no. IRT1132), and National Fund for Fostering Talents of basic Science (no. J1103202). Parts of calculations were performed at the High Performance Computing Center (HPCC) of Jilin University.

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