Superconductivity and unexpected chemistry of germanium hydrides under pressure

TL;DR

This study predicts new stable germanium hydrides under high pressure, revealing their structures, stability, and superconducting properties, and highlights their unique phase behavior compared to similar elements.

Contribution

It introduces a new stable germanium hydride structure with superconductivity, expanding understanding of high-pressure hydrides and their phase stability.

Findings

New C2/m germanium hydride stable above 278 GPa

Superconductivity with T$_c$ = 67 K at 280 GPa

Decomposition to Ge$_3$H$_{11}$ with T$_c$ = 43 K above 285 GPa

Abstract

Following the idea that hydrogen-rich compounds might be high-T$_c$ superconductors at high pressures, and the very recent breakthrough in predicting and synthesizing hydrogen sulfide with record-high T$_c$ = 203 K, ab initio evolutionary algorithm for crystal structure prediction was employed to find stable germanium hydrides. In addition to the earlier structure of germane with space group Ama2, we propose a new C2/m structure, which is energetically more favorable at pressures above 278 GPa (with inclusion of zero point energy). Our calculations indicate metallicity of the new C2/m phase of germane with T$_c$ = 67 K at 280 GPa. Germane is found to exhibit thermodynamic instability to decomposition to hydrogen and the new compound Ge$_3$H$_{11}$ at pressures above 300 GPa. Ge$_3$H$_{11}$ with space group I$\bar{4}$m2 is found to become stable at above 285 GPa with T$_c$ = 43 K. We find that the pressure-induced phase stability of germanium hydrides is distinct from its analogous isoelectronic systems, e.g., Si-hydrides and Sn-hydrides. Superconductivity stems from large electron-phonon coupling associated with the wagging, bending and stretching intermediate-frequency modes derived mainly from hydrogen.

Tables

Table 1. Table 1: Predicted crystal structures of Ge 3 H 11 and GeH 4 at 300 GPa.

Phase	Lattice	Atom	x	y	z
	parameters
I$\overline{4}$m2-Ge3H11	a=2.891 Å	Ge1(4e)	0.0000	0.0000	0.1750
	c=9.845 Å	Ge2(2b)	0.0000	0.0000	0.5000
		H1(8i)	0.2248	0.0000	0.3320
		H2(8i)	0.7377	0.0000	0.0351
		H3(4f)	0.0000	0.5000	0.1031
		H4(2c)	0.0000	0.5000	0.2500
C2/m-GeH4	a=10.226 Å	Ge1(2b)	0.0000	0.5000	0.0000
	b=2.967 Å	Ge2(4i)	0.8483	0.0000	0.6037
	c=2.922 Å	H1(8j)	0.3501	0.2383	0.1187
	$\beta$=74.46∘	H2(4i)	0.2822	0.0000	0.9765
		H3(4i)	0.2806	0.0000	0.6187
		H4(4i)	0.4274	0.0000	0.5731
		H5(4i)	0.9953	0.0000	0.7488

Table 2. Table 2: The calculated EPC parameter ( λ 𝜆 \lambda ), logarithmic average phonon frequency ( ω l ​ o ​ g subscript 𝜔 𝑙 𝑜 𝑔 \omega_{log} ) and critical temperature ( T c ) (with μ ∗ superscript 𝜇 \mu^{*} = 0.10 and 0.13) for C2/m -GeH 4 and I 4 ¯ ¯ 4 \bar{4} m2 -Ge 3 H 11 at given pressures.

Structure	Pressure (GPa)	$\lambda$	${\omega }_{log}$ (K)	Tc (K)
C2/m-GeH4	280	0.895	1162	67 (${\mu }^{\ast }$=0.10)
				56 (${\mu }^{\ast }$=0.13)
	300	0.867	1154	63 (${\mu }^{\ast }$=0.10)
				52 (${\mu }^{\ast }$=0.13)
I$\overline{4}$m2-Ge3H11	285	0.721	1155	43 (${\mu }^{\ast }$=0.10)
				34 (${\mu }^{\ast }$=0.13)
	300	0.690	1140	38 (${\mu }^{\ast }$=0.10)
				29 (${\mu }^{\ast }$=0.13)
	320	0.668	1127	35 (${\mu }^{\ast }$=0.10)
				26 (${\mu }^{\ast }$=0.13)

Full text

Superconductivity, hydrides, high pressure, evolutionary algorithm, density functional theory

High-throughput materials discovery using first-principles density functional theory (DFT) Kohn and Sham (1965) has motivated many experimental studies. For years, scientists have been trying to find the best way to design high-temperature superconductors. It has been confirmed that high-Tc superconductivity can be found in systems with light elements. Hydrogen is the lightest element with rich structures and properties under high pressures. Within BCS (Bardeen-Cooper-Schrieffer) theory of superconductivity Bardeen et al. (1957), high vibrational frequencies of hydrogen atoms and often high electron-phonon coupling make it possible to expect high-Tc in metallic hydrogen and hydrogen-rich hydrides.

However, metallic hydrogen seems to require very high pressure $\sim$ 400 GPa and proved elusive. Therefore, chemical precompression by alloying with heavy element was proposed Ashcroft (2004). Many theoretical and experimental studies have been motivated by this idea to seek and design new high-Tc superconductors at high pressures Gao et al. (2008a); Zhong et al. (2016); Errea et al. (2016); Zhong et al. (2012); Zhou et al. (2011); Hou et al. (2015); Duan et al. (2015a); Struzhkin (2015); Eremets, M. I. and Trojan, I. A. and Medvedev, S. A. and Tse, J. S. and Yao, Y. (2008); Drozdov et al. (2015a, b).

In a recent breakthrough discovery, which was first predicted by the evolutionary algorithm USPEX coupled with DFT studies Duan et al. (2014), high-temperature superconductivity with a transition temperature (Tc) of 203 K in hydrogen sulfide H3S under pressure 200 GPa has been reported by Drozdov et al. Drozdov et al. (2015b). This discovery not only set a record high-Tc for a conventional phonon-mediated mechanism, but also raised hopes of reaching room-temperature superconductivity in hydrogen-rich metallic alloys. This realization is the best argument to show the predictive power of DFT-based structure prediction and electron-phonon coupling calculations, and opens up avenues for discovering superconductors based on this approach.

Successful synthesis of hydrogen sulfides with superconducting properties was followed by a second high-Tc hydrogen-rich compound at high pressure (PH3) synthesized by Drozdov et al.Drozdov et al. (2015a). Prior to H3S, the highest experimentally observed Tc in conventional superconductors, which obey the BCS theory was in MgB2, which opened avenues for searching for higher Tc superconductors. However, other magnesium borides MgxBy were shown to exhibit poor superconductivity with Tc $<$ 3 K. Besides these efforts, other superconductors have been predicted in hydrogen-rich compounds. In group-IV hydrides, SiH4 has been predicted to have Tc = 20-75 K Chen et al. (2008), while experiment got a lower value of 17 K Eremets, M. I. and Trojan, I. A. and Medvedev, S. A. and Tse, J. S. and Yao, Y. (2008). Disilane (SiH8) has been predicted to favor Ccca structure with Tc of 98-107 K at 250 GPa Strobel et al. (2009). Our work on tin hydrides showed rich chemistry of that system with high-Tc superconductivity. Tin hydrides have been predicted to form at high pressures, exhibiting high Tc of 81, 93, 97 K for SnH8, SnH12 and SnH14 at 220, 250 and 300 GPa, respectively Esfahani et al. (2016). In addition, novel linear and bent formation of H3 and H4 have been predicted to form in high-pressure phases of SnH8, SnH12 and SnH14 Esfahani et al. (2016).

Germanium (Ge) is in the same group-IV and is isovalent to Sn. One can expect germanium to exhibit similar chemistry as tin, but its smaller atomic radius and slightly higher electronegativity than Sn result in quite a different chemistry.

Germane (GeH4) phases have been explored by Gao et al. Gao et al. (2008a) and their results show C2/c-GeH4 becomes stable at pressures above 196 GPa (including zero point energy (ZPE)) against decomposition into H and Ge. However, stability against decomposition into the elements is not a particularly stringent test, and stability against separation into other phases, e.g., GeH4 into Ge2H and H2, which is important for understanding decomposition mechanism, should be taken into account. C2/c was predicted to be a superconductor with Tc = 64 K at 220 GPa. In a recent theoretical study, a more energetically stable structure of germane (with symmetry group Ama2) was predicted by Zhang et al. to have Tc of 47-57 K Zhang et al. (2015). Now, with major progress of computational methods (enabling, for example, variable-composition searches), we can address all the outstanding issues.

We systematically explored the high-pressure phase diagram of Ge-H system with using evolutionary variable-composition search implemented in the USPEX code Oganov et al. (2011, 2010); Glass et al. (2006); Lyakhov et al. (2010) from ambient pressure to 400 GPa. The effectiveness of this method has been shown by the prediction of high-pressure structures of various systems that were subsequently confirmed experimentally (e.g., Zhang et al. (2013); Oganov et al. (2009); Mannix et al. (2015)). In this method, we created initial generation of structures and compositions using the random symmetric algorithm Lyakhov et al. (2013). Subsequent generations were obtained using heredity, transmutation, softmutation, and random symmetric generator Lyakhov et al. (2013). Ge hydrides, in comparison with other hydrides of the same group, e.g., Si Strobel et al. (2009); Chen et al. (2008), Sn Esfahani et al. (2016) which often show simpler phase diagram, exhibit a unique and complex potentianl energy landscape. Unexpected stoichiometries Ge3H, Ge2H, GeH3, GeH4 and Ge3H11 emerge as stable at megabar pressures.

The underlying structure relaxations were carried out using VASP package Kresse and Furthmüller (1996) in the framework of density functional theory (DFT) and using PBE-GGA (Perdew-Burke-Ernzerhof generalized gradient approximation) Perdew et al. (1996). The projector-augmented wave approach (PAW) Blöchl (1994) was used to describe the core electrons and their effects on valence orbitals. A plane wave kinetic energy cutoff of 1000 eV for hard PAW potentials and dense Monkhorst-Pack k-points grids with reciprocal space resolution 2$\pi$ $\times$ 0.03 Å*-1* were employed Monkhorst and Pack (1976) to sample the Brillouin zone.

Phonon frequencies and superconducting properties were calculated using density functional perturbation theory as implemented in Quantum ESPRESSO package Giannozzi et al. (2009). PBE-GGA functional is used for this part. A plane-wave basis set with a cutoff of 80 Ry gave a convergence in energy with a precision of 1 meV/atom. We used valence electron configurations of 3d10 4s2 4p2 and 1s1 for germanium and hydrogen, respectively. Thermodynamic properties of germanium hydrides were calculated using the PHONOPY package with the implemented frozen-phonon approach Togo and Tanaka (2015).

The electron-phonon coupling (EPC) parameter $\lambda$ was calculated using 5$\times$5$\times$2 and 4$\times$4$\times$4 q-point meshes for I$\bar{4}$m2-Ge3H11 and C2/m-GeH4, respectively. Denser k-point meshes, 20$\times$20$\times$8 and 16$\times$16$\times$16 were used in the calculations of the electron-phonon interaction matrix elements. The superconducting Tc, was estimated using the Allen-Dynes modified McMillan equation Allen and Dynes (1975).

The energetic stability of a variety of GexHy (x + y $<$ 20) compounds was evaluated using the thermodynamic convex hull construction at different pressures (see Supplemental Material Fig. S1). To our surprise, in addition to reproducing various structures of Ge-H system Hou et al. (2015); Gao et al. (2008a); Zhang et al. (2015, 2010), Ge Ribeiro and Cohen (2000) and H2 Pickard and Needs (2007), previously unreported and unexpected composition of germanium hydrides Ge3H11 was found to be stable in wide pressure ranges.

Below 200 GPa, no hydrogen-rich composition is stable against decomposition into the elements. It is consistent with not having any solid H-rich Ge-hydrides at low pressures, although using in situ gas-condensation techniques, Maley et al. showed germane can form at ambient pressure Maley et al. (2008). Increasing pressure decreases formation enthalpies, implying a tendency for Ge-hydrides to be stabilized under further compression. Phases of elemental H and Ge for the convex hull construction were obtained from structure search, in good agreement with the ones reported in Ref Pickard and Needs (2007) for hydrogen and for elemental Ge, we obtained a complex phase diagram with at least four phase transitions between 70 and 400 GPa, which are in good agreement with Ref Ribeiro and Cohen (2000).

At 250 GPa, the tetragonal Ge3H11 with space group I$\bar{4}$m2 is still metastable and lies just above the tie-line joining Ama2-GeH4 and Pnma-Ge2H. At 300 GPa, we predict stable phases: Ge3H (P63/m), Ge2H (Pnma) and GeH3 (Cccm) in accord with previous predictions Hou et al. (2015); Zhang et al. (2015). In addition, we also found unexpected composition Ge3H11 that appears in the H-rich region, its structure featuring GeH12 distorted icosahedra and GeH16 Frank-Casper polyhedra. Moreover, germane transforms to a new monoclinic phase with space group C2/m with 3 f.u./cell at above 300 GPa (278 GPa with inclusion of ZPE), which is lower in enthalpy than all previously proposed structures Gao et al. (2008a); Hou et al. (2015); Zhang et al. (2015) (see also Fig. 1(b)).

The stability fields of solids Ge3H, Ge2H, GeH3, Ge3H11 and GeH4 are illustrated in pressure-composition phase diagram of the Ge-H system, as shown in Fig. 1(a). Ge-rich compounds tend to stabilize at lower pressure ($<$200 GPa), while higher pressure ($>$200 GPa) is required for H-rich compounds to form. To the best of our knowledge, these unexpected, yet complex stoichiometries have not been reported in group-IV hydrides except MH4 (M = Si, Sn, Pb). This rich chemistry makes Ge hydrides of special interests. It can be seen that the formation of Ge3H11 at 285 GPa lowers the convex hull and finally around 300 GPa causes GeH4 to become thermodynamically metastable. The dynamical stability of structures shown in Fig. 1(a). were confirmed in their pressure ranges of stability via phonon calculations.

GeH4 was predicted to become stable against decomposition into the elements at above 225 GPa (196 GPa with the inclusion of zero-point energy) Gao et al. (2008b), while our results reveal lower enthalpy of Ge2H+H2 indicating the need for somewhat higher pressure 244 GPa (216 GPa with ZPE inclusion) for GeH4 to be stabilized (see Fig. 1(b)-inset). Upon increasing pressure, the Ama2 structure of GeH4 transforms into the C2/m structure at 300 GPa. Structures predicted in the literature are also included for comparison. In the Ama2 $\rightarrow$ C2/m transition, the coordination number of Ge atoms increases from 10 to 12 and 16 with the formation of GeH12 distorted icosahedra and GeH16 Frank-Casper polyhedra at 300 GPa (Fig. 2(a)-inset). In addition, the average Ge-H bond lengths slightly increase from 1.698 Å to 1.704 Å in the Ama2 $\rightarrow$ C2/m transition.

GeH4 is unstable against decomposition to H2 (Cmca) and Ge3H11 (I$\bar{4}$m2) at pressures above 300 GPa, according to the convex hull (see Supplemental Material Fig. S1(b) and (c)). Similarly, GeH3 decomposes to Ge2H and Ge3H11.

Both in I$\bar{4}$m2-Ge3H11 and C2/m-GeH4, each Ge atom is coordinated with 12 and 16 H atoms making distorted icosahedra and GeH16 Frank-Casper polyhedra (see Fig. 2). The average Ge-H bond lengths are 1.660 and 1.704 Å in I$\bar{4}$m2-Ge3H11 and C2/m-GeH4 at 300 GPa, respectively. Unlike other compressed hydrides Zhong et al. (2016); Esfahani et al. (2016); Duan et al. (2015b); Hooper et al. (2013), there are no bonds between H atoms.

As shown in the Fig. 2., liberating one hydrogen atom from a 3 f.u. cell turns a GeH16 polyhedra to a less coordinated germanium atom and leads to the formation of a distorted icosahedron, i.e., GeH4 consists of two GeH16 polyhedra and a GeH12 icosahedron, however, Ge3H11 turns out to have one GeH16 polyhedron and two distorted icosahedra. The detailed crystallographic data are listed in Table I.

Because of high concentration of hydrogen in GeH4, contribution of zero-point energy (ZPE) would be important in determining the relative stability of hydrogen-rich phases Shamp et al. (2016); Wang et al. (2012); Gao et al. (2008b); Errea et al. (2016); Esfahani et al. (2016). However, our results show that ZPE does not change the topology of the phase diagram of GeH4, and quantitative effects are just moderate shifts in transition pressures. For example, the inclusion of ZPE lowers the formation enthalpies of Ama2 and C2/m structures and shifts the transition pressure Ama2 $\rightarrow$ C2/m from 300 to 278 GPa indicating enhanced stability of C2/m phase owing to ZPE (see Fig. 1(b)-inset).

Analyzing the electronic band structures of GeH4 (C2/m) and Ge3H11 (I$\bar{4}$m2) (see Supplemental Material Fig. S2(a) and (b)) indicates indirect band overlap which results in metallic behavior, with highly dispersive bands crossing the Fermi level, these bands being basically due to germanium states with p-character and marginally due to hydrogen states with s-character. These H-derived states near Fermi level resemble those of solid metallic hydrogen. The C2/m structure is a metal with parabolic dispersive bands crossing the Fermi level along the A-$\Gamma$-Z symmetric line with several electron and hole pockets at the Fermi level. In the energy region near Ef, the DOS of Ge is about two times that of H, which indicates the dominance of Ge atoms contribution in the bands near the Fermi level. The total DOS at Ef, N(Ef), is 0.27 states/eV/f.u. for the C2/m-GeH4 structure at 300 GPa, while we see higher N(Ef) = 0.31 for Ama2 phase at 300 GPa. The Fermi levels of GeH4 and Ge3H11 fall on a shoulder of the density of states, while the record Tc in H3S is explained to be due to the van Hove singularity close to the Fermi level Quan and Pickett (2016); Ortenzi et al. (2016), therefore, doping can be expected to raise N(Ef) and Tc values. These values of DOS at the Fermi level N(Ef) are lower than those in H3S (0.54 states/eV/f.u.).

To probe the possible superconducting behavior, electron-phonon coupling (EPC) calculations were performed for C2/m-GeH4 and I$\bar{4}$m2-Ge3H11 structures at 280, 300 and 320 GPa. Phonon dispersions, phonon density of states, the corresponding Eliashberg spectral function $\alpha^{2}F$($\omega$) and the EPC parameter $\lambda$ as a function of frequency are calculated and shown in Fig. 3(a) and (b) for C2/m-GeH4 and I$\bar{4}$m2-Ge3H11 at 300 GPa, respectively.

The low frequency bands below 430 cm*-1* are mainly from the strongly coupled vibrations between Ge and H that contribute about 26% (25%) of the total $\lambda$, while higher-frequency phonons, predominantly wagging, bending and stretching modes between 550-2300 cm*-1* are mostly related to the H atoms bonded to Ge and contribute 74% (75%) of $\lambda$ of C2/m-GeH4 (I$\bar{4}$m2-Ge3H11) phase.

The resulting integral $\lambda$ and logarithmic average phonon frequencies ($\omega_{log}$) are calculated using the Eliashberg formalism and then Tc values are estimated using Allen-Dynes modified McMillan equation with using Coulomb pseudopotential parameters $\mu^{*}$ = 0.1 and 0.13 as commonly accepted values. Table II summarizes data for the total EPC parameters $\lambda$, logarithmic phonon average frequencies and corresponding Tc values at given pressures.

Hard phonons in H-rich materials are described to play an important role in high-Tc superconductivity Ashcroft (1968), but because such hard phonons do not always produce large coupling constant, high-Tc superconductivity is still elusive. In the C2/m-GeH4 structure, high-frequency vibrations that contribute the most to EPC parameter, produce a larger coupling constant i.e., 25% higher than similar frequency modes in I$\bar{4}$m2-Ge3H11. Additional flat bands in the high frequency region of C2/m-GeH4 phonon modes, can be ascribed to the higher coupling constant and eventually result in getting higher Tc value for C2/m-GeH4.

We investigated the pressure dependence of the critical transition temperature. The results show that the calculated Tc decreases monotonically with pressure with approximate rates of -0.19 and -0.20 K/GPa for C2/m-GeH4 and I$\bar{4}$m2-Ge3H11 in the pressure range 280-320 GPa. Higher Tc of the C2/m phase, compared to the previously reported phase Ama2-GeH4, can be related to the considerably higher average phonon frequency, which also can be explained through a BCS mechanism.

In summary, we explored high-pressure phase diagram of the Ge-H binary system by exploring its compositional and configurational space with an evolutionary crystal structure prediction method. Based on analysis of current and prior theoretical studies on Ge-hydrides, we have established thermodynamically stable phases, superconducting properties, structural features and new decomposition lines in the pressure range 0-400 GPa.

At 250 GPa, all the stoichiometries Ge2H, Ge3H and GeH4 are energetically stable against any decomposition into the elements or any other compounds. At 300 GPa, GeH3 and Ge3H11 become stable, while GeH4 becomes unstable.

A unique metallic phase of germane with C2/m space group is found to be energetically more favorable than all previously proposed structures at pressures above 278 GPa (if zero-point energy is included). Our results reveal that germane decomposes to hydrogen and the newly found compound Ge3H11 at the pressures above 300 GPa. According to electron-phonon coupling calculations, C2/m-GeH4 and I$\bar{4}$m2-Ge3H11 are excellent superconductors with high-Tc of 67 K and 43 K for C2/m-GeH4 at 280 GPa and I$\bar{4}$m2-Ge3H11 at 285 GPa, respectively.

I Acknowledgements

We thank DARPA (grant W31P4Q1210008) and the Russian Science Foundation (grant 16-13-10459) for financial support. Calculations were performed on XSEDE facilities and on the cluster of the Center for Functional Nanomaterials, Brookhaven National Laboratory.