Prediction of thermodynamic stability and electronic structure of novel ternary lanthanide hydrides

Abstract

We theoretically examine four hypothetical ternary lanthanide hydrides, CsLn^IIH₃ and Cs₂Ln^IIH₄, where Ln^II = Yb, Tm. We optimize their crystal unit cells in the BaTiO₃ and K₂NiF₄ structures, respectively, and compute their electronic band structures. Our calculations indicate that the novel hydrides should be unstable with respect to decomposition into binaries (CsH and Ln^IIH₂); ternaries would form only under elevated pressure (>7–22 GPa). We predict that significant perturbation of the electronic and magnetic properties of CsYb^IIH₃ and Cs₂Yb^IIH₄ will take place via a progressive exchange of f¹⁴ Yb^II for f¹³ Tm^II, while promoting magnetic ordering and valence fluctuations. Analysis of the phonon dispersion for these hydrides suggests that metallic forms of doped CsLn^IIH₃ and Cs₂Ln^IIH₄ would exhibit substantially high Debye temperatures of ∼1800 K. This is likely to prompt moderate-T_C superconductivity in these as yet unknown materials.

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Introduction

Among lanthanide metals, the divalent state is favored predominantly for Yb and Eu, due to the stability of—respectively—filled (f¹⁴) and half-filled (f⁷) f shell. Significantly lesser stability is noticed for the elements preceding Yb and Eu in the periodic table, namely Tm and Sm, respectively.¹ This is attested to by the standard redox potentials for the M^III/M^II redox pairs (E⁰ = −1.05 V for Yb, while −2.30 V for Tm; −0.35 V for Eu, while −1.55 V for Sm).² As a consequence, the nearly stoichiometric trihydrides of trivalent Tm and Sm are known, while genuine YbH₃ and EuH₃ have never been synthesized. For Yb only the mixed-valence Yb^II(Yb^IIIH₄)₂ has been obtained so far,³ while for Eu, EuH_1.95 is a limiting composition, in agreement with the ranking of the E⁰ values and other thermodynamic arguments.⁴

Yb^II and Eu^II, the most prominent representatives of stable divalent lanthanide cations, are often compared to closed shell Ca^II and Sr^II, respectively, in terms of ionic radius.⁵ A nice series of isomorphic compounds of Yb^II and Ca^II, and those of Eu^II and Sr^II, are known; examples include YbBeF₄ and CaBeF₄, EuLiF₃ and SrLiF₃, and—among the hydrides—Yb₄Mg₃H₁₄⁶ and Yb₄Mg₄Co₃H₁₉⁷ (and their Ca analogues), Eu₆Mg₇H₂₆⁸ and inverse perovskite EuLiH₃ (and their Sr analogues). In all these compounds, binary components of divalent Yb and Eu (i.e., EuF₂, YbH₂etc.) formally serve as Lewis bases, while donating their ligands to the stronger and smaller Lewis acids (Mg^II, Be^II).

Tm^II (ionic radius, R, of 1.17 Å in the octahedral environment), an electron-deficient analogue of Yb^II, is virtually isomorphic with Yb^II (R = 1.16 Å)⁹ and with Ca^II (R = 1.14 Å). However, because of the enormous reducing properties of Tm^II,¹⁰ its compounds are scarce, and very difficult to preserve in the presence of traces of O₂ or humidity.

Prompted by the existence of M^ICaX₃ and M^I₂CaX₄, where M^I = Cs, Rb and X = H or F,¹¹ of ternary lanthanide salts such as the perovskites CsYbF₃,¹² CsYbI₃ and RbTmI₃,¹³ and driven by the fluoride-hydride analogy,¹⁴ we examine theoretically and attempt to predict the possible existence and selected properties of CsLn^IIH₃ and Cs₂Ln^IIH₄ where Ln^II = Yb, Tm. In these hypothetical compounds, and in contrast to all known ternary hydrides containing divalent lanthanide cations, LnH₂ would formally serve as a Lewis acid while accepting hydride anion from the very strong base (CsH). Here, our interest in ternary hydrides of divalent lanthanides has origin in (i) anticipated involvement of f orbitals in bonding, i.e. a mixing of Ln^II (4f) and H (1s) orbitals (moderate ‘covalence’), (ii) large values of electronic density of 4f states in the vicinity of the Fermi level, and (iii) concomitant consequences of hole- and electron-doping for electronic and magnetic properties of the stoichiometric compounds of Ln^II, and possible promotion of superconductivity.

Methods of calculations

Our computations were based on density functional theory and used a generalized gradient approximation (GGA) with ultrasoft Vanderbilt-type pseudopotentials, as implemented in the CASTEP¹⁵ code. For each structure, we relaxed the unit cell constants and atomic positions (whenever possible), and also determined the stress tensor. The electronic density of states, band structure and electronic density within a desired energy window were subsequently calculated for the optimized unit cell. The k-point grids for zone sampling were generated automatically via the Monkhorst–Pack scheme. The k-point sampling used by CASTEP was 4 × 4 × 4 (structures of CsLnH₃), and 4 × 4 × 2 (Cs₂LnH₄). We used a 450 eV cutoff for the kinetic energy of the plane waves and an SCF tolerance of 2 × 10⁻⁶ eV per atom. In our calculations we used plane waves originating from an expansion of four valence and sub-valence orbitals for Cs (5s², 5p⁶, 6s², 6p⁶), two orbitals for each H (1s, 2s) and six orbitals for Yb and Tm (5s², 5p⁶, 4f¹⁴, 6s², 6p², 5d¹⁰). This gives a total of 31 bands for CsYbH₃ (Z = 1); 18 of these are occupied, 13 are unoccupied. No other atomic functions (in particular no polarization functions on H, 2p) are embedded in the CASTEP package for these atoms.

For compounds of Tm^II (an open-shell f¹³ configuration with large magnetic moment) we have always done spin-polarized calculations (SP). Calculations without spin polarization (‘metallic’ or ‘antiferromagnetic’ polymorph) always yielded a larger energy per unit cell (by about 0.5 eV per Tm atom) than for those with polarized spins (‘ferromagnetic’ polymorph). At the same time, calculations without spin polarization typically result in minor changes in the unit cell vectors (for example, compare results for Cs₂TmH₄: SP a = b = 4.601 Å, c = 15.696 Å, no SP a = b = 4.616 Å, c = 15.613 Å).

We have also performed supplementary VASP¹⁶ calculations for several Yb- and Tm-containing compounds (see the electronic supplementary information (ESI)†). Here we could use dense k-point sampling of 11 × 11 × 11 (CsLnH₃), and 11 × 11 × 6 (Cs₂LnH₄). We tested two types of pseudopotentials: those which explicitly involve f electrons on Ln atoms (PBA set) and those which do not; it turned out that optimizations with VASP (pseudopotentials without f electrons) and CASTEP (pseudopotentials with f electrons) yielded only minor differences in unit cell vectors for CsYbH₃ and Cs₂YbH₄.

For phonon dispersion and phonon DOS calculations we have used the PHONON¹⁷ code as implemented in the MEDEA package. PHONON uses VASP-preoptimized unit cells in order to determine vibrations of the crystal lattice. To speed up the time-demanding frequency calculations we used norm-conserving pseudopotentials which do not explicitly involve f electrons on Ln atoms.

Results and discussion

Crystal structures

Due to the Ca^II–Ln^II (Ln = Yb, Tm) isomorphism mentioned above, one may expect that CsLn^IIH₃ and Cs₂Ln^IIH₄ will adopt, respectively, the Cs^ICaH₃ (perovskite) and Cs^I₂CaH₄ (K₂NiF₄) structures. In Table 1 we present the results of unit cell optimization for four novel ternary lanthanide hydrides, and compare them with the experimental and computational data for analogous calcium salts.¹¹

Table 1 The optimized unit cell vectors for CsMH₃ and Cs₂MH₄ where M = Yb, Tm, Ca, and the corresponding experimental values for Ca salts (CASTEP results). Fractional z atomic coordinate for an apical H_(0,0,z) atom is given in bracket

Compound	CsYbH3	CsTmH3	CsCaH3	CsCaD3 exp.	Cs2YbH4	Cs2TmH4	Cs2CaH4	Cs2CaD4 exp.
a = b/Å	4.609	4.609	4.619	4.617	4.609	4.601	4.615	4.597
c/Å	4.609	4.609	4.619	4.617	15.531 (0.148)	15.696 (0.146)	15.558 (0.149)	15.528 (0.149)

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Our GGA-PBA calculations predict a cubic unit cell constant of 4.609 Å for CsYbH₃ and 4.609 Å as well for CsTmH₃; the deviation from the experimental value for CsCaD₃ (4.617 Å) is very small (<0.01 Å) and slightly negative; in fact small positive deviations (ca. +0.03 to 0.04 Å) are expected based on comparison of the crystal radii (see the previous section). The difference between calculated and experimental cubic unit cell constants for CsCaD₃ is as small as 0.002 Å.

Our result is entirely within the limitations of the computational method utilized (we note slightly positive deviations for smaller energy cutoffs). The calculated unit cell constant of 4.609 Å for CsYbH₃ also matches well the experimental value of 4.61 Å for CsYbF₃; indeed, the close hydride–fluoride analogy has been pointed out to apply for similar compounds.¹⁴ For CsTmH₃, the environment of Tm^II is predicted to be that of an ideal octahedron; the Jahn–Teller (JT) distortion apparently does not influence the coordination shell of the f¹³ cation, similar to the crystal structure of CsTmCl₃.¹⁸ This feature points to the relatively ionic character of Tm–H bonding.

For Cs₂YbH₄, slight deviations from regular octahedral coordination of Yb^II are predicted, with four equivalent longer equatorial Ln–H distances of 2.305 Å and two shorter apical distances of 2.301 Å. This effect is of course slightly larger for ferromagnetic (f¹³) Cs₂TmH₄ (equatorial 2.300 Å, apical 2.286 Å). An inverse JT effect is computed to be small for the inner transition metal f¹³ configuration of Tm^II (the ratio of apical to equatorial Tm–H distances is 0.994), in agreement with experimental data for similar compounds of Tm^II (the analogous ratio being 0.986 for KTmI₃,¹⁹ and 1.006 for Cs₂TmCl₄²⁰).

The fractional z atomic coordinate for an apical H_(0,0,z) atom (which quantitatively determines the distortion of the {LnH₆} octahedron), has been predicted to be 0.146 for Cs₂TmH₄, via 0.148 for Cs₂YbH₄, up to 0.149 for Cs₂CaH₄, always close to the experimental value of 0.149 for Cs₂CaD₄.

The predicted unit cell vectors are very similar for hydrides of Yb^II and for isostructural compounds of Tm^II (identical for CsLnH₃, and within 1% deviation for Cs₂LnH₄). This should facilitate formation of solid solutions for analogous compounds of these Ln metals, i.e., ease of electronic doping.

Thermodynamic stability

In Table 2 we present the calculated enthalpies of formation²¹ (at T = 0) of the ternary caesium lanthanide hydrides studied, with respect to formation from CsH and LnH₂ (CsH and LnH₂ have been optimized based on the experimental crystallographic data²²).

Table 2 The calculated enthalpies of formation (ΔH), volume changes (ΔV) and estimated formation pressures, p, for the formation reaction from binaries for CsMH₃ and Cs₂MH₄ where M = Yb, Tm

Compound	CsYbH3	CsTmH3	Cs2YbH4	Cs2TmH4
ΔH/eV per molecule	0.27	0.55	0.19	0.47
ΔV/Å^3 per molecule	−6.02	−5.30	−5.66	−3.85
p/GPa	7.3	16.7	5.3	19.7

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Ternary Tm and Yb hydrides are predicted to be thermodynamically unstable with respect to decomposition to binary hydrides, and it is not expected that vibrational zero-point correction and entropy effects—typically small for solid reagents—will easily reverse this trend. Diamagnetic forms of Tm^II compounds are computed to have higher electronic energy than their magnetically ordered (ferromagnetic) polymorphs, in agreement with the strong magnetism of unpaired f electrons. In the following sections we will discuss exclusively ferromagnetic compounds of Tm^II.

The enthalpy of the reactions of synthesis of ternary hydrides is positive for all ternaries studied, while the associated volume change is computed to be negative (Table 2). This suggests that hydrides might form from binaries at elevated pressure. A rough estimate of the formation pressure can be obtained from the “common tangent” method, as a ratio of reaction enthalpy to the volume change. The estimated formation pressures—listed in Table 2—range from 5 to 20 GPa.²³

Our conclusion is that ternary hydrides of Tm^II and Yb^II could be synthesized at increased pressures, and at relatively low temperatures up to 200 °C (to avoid decomposition of fragile CsH) and with rather small H₂ overpressure (to avoid formation of compounds of trivalent lanthanides, especially for compounds of divalent Tm, strong reducing agents). Some of these ternary phases might be quenchable upon decompression to 1 atm.

Electronic structure

Let us now analyze the electronic structure of CsLnH₃ and Cs₂LnH₄.

The band structure of Cs/Yb(Tm) hydrides (see ESI†) consists of: (i) very narrow filled subvalence bands originating from 5s² (at about −52 eV) and 5p⁶ (at about −24 eV) states of Yb(Tm), and from 5s² (at about −23 eV) and 5p⁶ (at about −10 eV) states of Cs, (ii) a filled ‘hydride band’ at −6 to −2.5 eV, (iii) a very narrow 4f valence band and (iv) a very broad set of bands constituting the conducting band. In the analysis to come we will omit the sub-valence bands, i.e. set (i).

The CASTEP software does not allow direct atomic identification of a given segment of the density of states, only decomposition by angular momentum type—s, p, d, or f. Nevertheless an atomic assignment can be made, both by the energy range in which the density is found, and also by computing the disposition in space of the states in a given energy interval.

The ‘hydride band’ (ii) is visibly split from the valence band (iii) for all ternary hydrides studied (by some 3 eV) and it is composed predominantly of H (1s) states (Fig. 1), with a little contribution from lanthanide s, p and d states; in turn, the valence 4f band contains very little contribution from H (1s) states, which might point to an ionic formulation of the hydrides studied, to be written formally as Cs^ILn^II(H⁻)₃ and (Cs^I)₂Ln^II(H⁻)₄. The ‘hydride band’ is usually split in several separate bands, some of which show significant dispersion (up to 3 eV). This cannot be due to direct H⋯H interactions, as the hydrides are far apart (usually at >3.2 Å). The band width in the hydride band is due to interactions between H ions and lanthanide and alkali centers; the title hydrides are not as ionic as one might think.

Fig. 1 Band structure and DOS for CsYbH₃ (upper left), CsTmH₃ (upper right), Cs₂YbH₄ (lower left) and Cs₂TmH₄ (lower right). Focus is on hydride, f and conduction bands.

The set of bands (iii) originating from 4f states either gives rise to a single and large peak in the DOS (as for diamagnetic f¹⁴ Yb^II) or is split equally into two bands separated by some 2 eV (as for ferromagnetic f¹³ Tm^II). This description of f electrons in these phases is unsatisfactory; unfortunately it is common to all first-principles calculations within the independent electron approximation.^24,25

The broad conduction band, or rather, the set of interpenetrating bands (with a collective width of up to 9.5 eV for CsLn^IIH₃ and of only 3 eV for Cs₂Ln^IIH₄) is built of s, p and d states, predominantly coming from Yb, but in some part also from the alkali metal. For Yb hydrides, the bottom of the (formally empty) conduction band is remarkably close (0.25–0.5 eV) to the uppermost part of the (formally filled) valence band. For Tm compounds, however, the broad conduction band penetrates through (and for CsTmH3 even 0.5 eV below) the narrow 4f band, thus closing the direct band gap.

The electronic density at the Fermi level consists mainly of rather localized 4f electrons (Fig. 2) with no contribution from H 1s states.

Fig. 2 Electronic density (in blue) at the Fermi level (E_F ± 0.03 eV) for CsYbH₃ (upper left), CsTmH₃ (upper right), Cs₂YbH₄ (lower left) and Cs₂TmH₄ (lower right). Density isovalue 0.012 e Å⁻³.

Phonon density of states

Prompted by our continuing search for novel superconducting materials (with particular emphasis on fluorides and hydrides),²⁶ and by an unprecedented dynamic scale of all H-containing solids, we have investigated the phonon dispersion and phonon density of states for CsYb^IIH₃ (Fig. 3).

Fig. 3 Phonon band structure (left) and phonon DOS (right) for CsYbH₃; PHONON/VASP calculation. H's contribution to phonon DOS is in black, Yb's one in red, Cs's one in blue.

The unit cell of CsYbH₃ contains 5 atoms, so there are 15 degrees of freedom: 12 vibrational and 3 translational ones. Three phonon bands are seen in the highest frequency range of the spectrum (30.9–37.7 THz, 128–156 meV, 1030–1260 cm⁻¹).²⁷ They are degenerate at the zone center (recollect, the cell is cubic), and correspond to the Yb–H stretching modes. Six other modes are seen in the moderate-frequency range (11.0–21.9 THz, 46–90 meV, 370–730 cm⁻¹). These are the Ln–H–Ln bending modes, which also show significant amplitude of H motions. The last set of three phonons is centered at low frequency (3.07 THz, 13 meV, 120 cm⁻¹). These are deformational modes of the lattice, with predominant contribution from heavy elements (Yb, Cs). The remaining three lowest-frequency motions correspond to translations of the entire lattice; their calculated wavenumbers (at the zone center, Γ) are close to zero within narrow error margin, as expected. The dispersion of Ln–H stretching modes is significant (∼7 THz, 29 meV, 230 cm⁻¹); that of Ln–H bending modes is even larger (∼10.8 THz, 45 meV, 360 cm⁻¹).

The phonon density of states is composed of a single narrow and intense peak originating from the lowest-frequency phonons; a low dispersion, broad band coming from moderate-frequency phonons; and a double peak of moderate intensity marking the highest energy range of the spectrum.²⁸

In the first section we have considered the ‘static’ crystal structure of lanthanide hydrides, and we have pointed to minor structural distortions predicted for the Cs₂Ln^IIH₄ compounds. We think that dynamic inversion of the Jahn–Teller effect (a set of local two short and four long Ln–H bonds transforms into two long and four short bonds) is undoubtedly within vibrational amplitude of H atoms, at least within the Born–Oppenheimer approximation. Interestingly, such fluctuating transformation of a JT species should decrease the anisotropy of electron density around the lanthanide cation, in particular for compounds of Tm^II.

Superconductivity in hydrides and valence fluctuations. Why is T_C for hydrides so low?

As one knows for many other systems, moderate and high-temperature superconductivity arises in the vicinity of the metal-to-insulator transition.²⁹ For mixed-valence systems, motions of ligands bridging two metal centers seem to be of crucial importance for phonon-driven superconductivity. Superconductivity has been detected for several hydride materials (for example Th₄H₁₅), yet despite the presence of a light H atom, the values of the transition temperature (T_C) are remarkably small (not exceeding 20 K). In addition, the isotope effect on the value of T_C is opposite to the predictions of the classical BCS theory³⁰ for Pd hydride, the best studied hydride superconductor.³¹

In a simplistic chemical picture, PdH_x≈1.0 is formally a mixed valence system, Pd⁰Pd^IIH⁻¹₂.³² A molecular mixed-valence systems can be either localized, delocalized (i.e., intermediate valence), or an intermediate case between the two.³³ These three classes of molecular mixed-valence systems may naturally be compared to, respectively, insulating, metallic and superconducting mixed-valence solids.³⁴ In this picture, the electronic wavefunction of a system (be it a solid or a molecule) is most affected by motions of nuclei when the vibrational zero-point energy (ZPE) of metal–ligand–metal stretching (or of the highest-energy optical phonon in solids) is close to the energy barrier for the electron transfer (E_B) between the metal centers (Fig. 4).

Fig. 4 The double-well PES for a mixed-valence compound for three different zero-point vibrational energy situations: (A) a separated (localized) mixed valence compound; (B) a fluctuating valence species; (C) an intermediate (delocalized) valence compound. Solid state analogues are likely to be (A) insulators, (B) superconductors, (C) ‘normal’ metals. Case (D) illustrates the postulated explanation for the inverse isotope effect observed experimentally for palladium hydride (PdH, PdD, PdT).

When the ZPE is too small, the system is localized (insulating, Fig. 4A). When the ZPE is too large, the system is delocalized (‘simple’ metal) and two distinct oxidation states cannot be recognized in the molecular (crystal) structure (Fig. 4C). We suggest that when the ZPE is close to E_B, strong electron–phonon coupling dramatically influences the shape of the wavefunction and superconductivity may arise (Fig. 4B).

For PdH, electron transfer between Pd centers is greatly affected by motions of the hydride ligand.³² Hydrogen is the lightest ligand available to chemistry; here the ZPE can reach as much as 0.25 eV (for molecular H₂). In cases when the ZPE for the proton isotope is larger than E_B, and since the (classical) amplitude of vibrations—and also the vibrational zero point energy—decreases as the atomic mass increases for three isotopes of H, inverse isotope effect might indeed occur (Fig. 4D).

The case of PdH (small T_C because of large energy of hydrogen motions (!), and concomitant easy mixing of d orbitals on Pd centers linked by H) brings to mind an idea for increasing the E_B value by (i) involving in the electronic transport such orbitals, which are deeper lying (in energy) and more contracted than the 4d set of Pd (in this contribution we want to use 4f electrons), or by (ii) increasing the separation of the metal centers.³⁵ It might allow a reversal of the atypical isotope effect, and also larger T_C values by hydride superconductors. CsTmH₃ and Cs₂TmH₄, which have a nearly filled yet significantly contracted f-shell, seem promising in terms of possible heavy-fermion superconductivity via appropriate chemical doping to these compounds; if E_B proves to be too large, external pressure might be applied to bring these systems back towards the metal–insulator transition.

Cs₂TmH₄ (f¹³ Tm^II) is a quite formal analogue of other important systems with a nearly filled electron shell, namely Ln₂CuO₄ and Nd₂CuO₄ (d⁹ Cu^II). The latter two are parent (undoped) compounds of, respectively, hole-doped (La_2−xBa_xCuO₄) and electron-doped (Nd_2−xCe_xCuO₄) oxocuprate superconductors. Simultaneously, CsTmH₃ perovskite (f¹³ Tm^II) may be treated as a distant sibling of BaBiO₃, with a distorted perovskite structure (s¹ Bi^IV). The latter becomes Ba_1−xK_xBiO₃, a regular perovskite superconductor, upon hole-doping.³⁶

Thus, in terms of a formal electron count on the transition metal, Cs₂Tm^II_1−xYb^II_xH₄ would be equivalent to Nd^III_2−xCe^IV_xCu^II/IO₄, while Cs_1−x□_xTm^II/IIIH₃ (where □ stands for a vacancy)³⁷ would be analogous to Ba_1−xK_xBi^IV/VO₃.³⁸

We expect that tuning the transition between the f¹³ and f¹⁴ electron count³⁹ within the LnH₂ sublattice may bring Cs₂Tm^II_1−xYb^II_xH₄ and CsTm^II_1−xYb^II_xH₃ towards the anticipated metal/insulator borderline. At this stage, it is impossible to evaluate correctly from first principles whether these quaternary lanthanide hydrides will exhibit genuine heavy-fermion superconductivity, or plain ‘valence fluctuations’ (commonly observed for compounds of Yb, Tm, Eu, Sm, and also Ce and Tb).^40,41 The consequences for the electronic and magnetic properties of novel hydride phases of the interplay of strong magnetism of 4f electrons and of the varying electron count (at the edge of the f shell closure), should be probed experimentally.

The estimated Debye temperature of CsYbH₃ is substantial, ∼1800 K, while DOS below the Fermi level rises very steep, due to narrowness of the 4f electron band. The BCS theory predicts that these features alone are likely to prompt low-T_C superconductivity in as yet unknown hole-doped hydrides of Yb^II, despite rather small involvement of H states into the DOS at the Fermi level of these compounds. Ultra-high pressure experiments constitute another prospective path of research. External pressure is an independent experimental variable which might enforce larger mixing of hydride 1s states into the quasi-valence 4f electrons and thus provoke larger involvement of the former in the electronic transport in these compounds. In this way a moderate-T_C superconductivity could be achieved while making use of strong electron–phonon coupling of the states at the Fermi level with the high-frequency H-based phonons.

Conclusions

We predict the existence and some properties of four novel stoichiometric ternary hydrides of Tm and Yb. Our calculations indicate that—under easily achievable pressure of ∼20 GPa—CsLnH₃, Ln = Yb, Tm, could be stable in the perovskite structure, while Cs₂LnH₄ might adopt the K₂NiF₄ one. These two structures are also shared by many 3D and 2D superconductors, and by some metallic hydrides.⁴² The electronic structure of these novel hydrides reveals a very narrow f band, very close to—or even interpenetrated by—the broad (s, p, d) conduction band. In addition, the phonon spectrum benefits from the high-frequency (up to 156 meV, 1800 K) modes involving motion of H atoms.⁴³ The crystal structures adopted by hydrides in question, the isostructurality of Yb^II and Tm^II, and the electron count in the vicinity of shell closure (f¹³–f¹⁴) create the potential for tuning of electronic and magnetic properties of novel phases in the vicinity of the metal-to-insulator transition. An alluring possibility of valence fluctuations, or even heavy fermion superconductivity, arises.

Acknowledgements

W.G. and T.J. want to thank the Interdisciplinary Center for Mathematical and Computational Modelling (ICM) at Warsaw University for providing access to its computational resources and financial support. The work at Cornell was supported by The Cornell Center for Materials Research and Cornell Summer Graduate Program. T.J. is grateful to the Ministry of Education and Sport for the Fellowship.

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22. E. Zintl and A. Harder, Z. Phys. Chem., Abt. B, 1931, 14, 265–284; C. E. Messer and P. C. Gianoukos, J. Less-Common Met., 1968, 15, 377–383; J. E. Bonnet and J. N. Daou, J. Appl. Phys., 1977, 48, 964–968; J. E. Bonnet and J. N. Daou, J. Phys. Chem., 1962, 66, 148–151.
23. The reaction path might be different for hydrides of Yb and for those of Tm. For Tm compounds, the pressure increase should lead first to CsTmH₃ (at 16.7 GPa) and then subsequently to Cs₂TmH₄ (at 19.7 GPa). For Yb compounds, however, CsYbH₃ will be difficult to reach, as formation of (Cs₂YbH₄ + YbH₂) is preferred over the whole pressure window studied. CsYbH₃ might possibly be obtained via controlled thermal decomposition of CsYb^IIIH₄.
24. N. W. Ashcroft and N. D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976, p. 308–309.
25. Modelling valence transitions in lanthanides from first principles is very difficult and resource-consuming. See for example: A. Delin, L. Fast, B. Johansson, J. M. Wills and O. Eriksson, Phys. Rev. Lett., 1997, 79, 4637–4640; W. M. Temmerman, Z. Szotek, A. Svane, P. Strange, H. Winter, A. Delin, B. Johansson, O. Eriksson, L. Fast and J. M. Wills, Phys. Rev. Lett., 1999, 83, 3900–3903.
26. W. Grochala and R. Hoffmann, Angew. Chem., Int. Ed., 2001, 40, 2743–2781; W. Grochala, J. Mol. Model., 2005, 11, 323–329; J. Feng, W. Grochala, T. Jaroń, R. Hoffmann, A. Bergara and N. W. Ashcroft, Phys. Rev. Lett. , in press; M. Martinez-Canales, A. Bergara, J. Feng and W. Grochala, J. Phys. Chem. Solids , in press.
27. The cut-off energy of the phonon spectrum (156 meV) is, of course, much smaller than that of the main vibron of solid H₂ (around 546 meV). This comes from the reduced mass factor (beneficial by ∼1.4 times for H₂) and from the smaller force constant for stretching of ionic Yb–H bonds than of strong H–H ones (by a factor of ∼2.5).
28. We have not calculated the phonon dispersion and phonon DOS for the other ternary hydrides in question (CsTmH₃, Cs₂YbH₄, Cs₂TmH₄). One can anticipate that these compounds will exhibit only small changes of phonon DOS as compared to CsYbH₃. For example, Cs₂YbH₄ has two sets of Yb–H bonds: four longer ones at 2.305 Å and two shorter ones at 2.301 Å, while CsYbH₃ has six equivalent Yb–H bonds at 2.305 Å. It means, that the Yb–H stretching mode (triply degenerate at Γ for cubic CsYbH₃) will now split into a doubly degenerate in-plane mode (of similar frequency as for CsYbH₃), and a non-degenerate mode of slightly higher frequency (connected with the Yb–H stretching in the crystallographic c direction). The former modes are responsible for the intra-layer electron–phonon coupling, and the latter mode for the inter-layer coupling. Additional minor splits are expected due to doubling of the Cs₂YbH₄ chemical formula in the unit cell of this compound (Z = 2).
29. R. S. Liu, P. P. Edwards, S. F. Hu, D. A. Jefferson, Y. T. Huang, S. F. Wu and M. J. Tsai, J. Electron. Mater., 1993, 22, 1199–1203; D. Meddeb, S. Charfi-Kaddour, M. Heritier and R. Bennaceur, Physica C, 2004, 408–410, 152–153; J. Halbritter, J. Supercond., 2003, 16, 833–841; M. S. Osofsky, R. J. Soulen, J. H. Claassen, G. Trotter, H. Kim and J. Horwitz, Phys. Rev. B, 2002, 66 Art. 020502 J. H. Jung, S. A. Seo, T. W. Noh, P. A. Sharma, N. Hur and S. W. Cheong, Solid State Commun., 2003, 126, 175–179.
30. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev., 1957, 108, 1175–1204.
31. T. Skośkiewicz, Phys. Status Solidi A, 1972, 11, K123 ; PdH_x shows inverse isotope effect: T_C = 8 K for protide, 10 K for deuteride, and 12 K for tritide: B. Stritzker, Z. Phys., 1972, 257, 1–8; J. E. Schirber, J. M. Mintz and W. Wall, Solid State Commun., 1984, 52, 837–838; H. Hemmes, A. Driessen, R. Griessen and M. Gupta, Phys. Rev. B, 1989, 39, 4110–4118; T. Skośkiewicz, A. W. Szafrański, W. Bujnowski and B. Baranowski, J. Phys. C, 1974, 7, 2670–2676 . It is argued that the influence of zero-point motions and anharmonicity accounts for this unusual behavior: M. Yussouff, B. K. Rao and P. Jena, Solid State Commun., 1995, 94, 549–553; B. M. Klein and R. E. Cohen, Phys. Rev. B, 1992, 45, 12405–12414.
32. Nonstoichiometric Pd hydride is very difficult to desribe using classical terms of chemistry such as cation and anion, or metal and ligand. In fact, some descriptions prefer the ‘solvated proton’ scenario, i.e., H⁺ is solvated by Pd^δ−. In chemical terms, the first ionization potential of Pd (804 kJ mol⁻¹) is smaller than that of H (1312 kJ mol⁻¹), while the first electron affinity of Pd (53.7 kJ mol⁻¹) is also smaller than that of H (72.8 kJ mol⁻¹). Both properties point rather to a Pd^δ+/H^δ− polarization in a palladium hydride, but the Pd–H bonding should of course be quite covalent.
33. M. B. Robin and P. Day, Adv. Inorg. Chem. Radiochem., 1967, 10, 247–422.
34. See ref. 26 and compare with: S. Larsson and A. Klimkãns, Int. J. Quantum Chem., 2000, 80, 713–720.
35. Note, conductivity and superconductivity usually requires presence of atomic bridges between the electron-exchanging centers in the crystal structure of an extended solid. Here we postulate that, paradoxically, certain molecular hydride materials might be even better superconductors than those hydrides which contain the direct M–H–M bridges.
36. The hydride ligand is unique among all inorganic ligands, as it has no p orbitals and therefore π bonding is not available. This is important in terms of hole-doping, and progressive depopulation of the 1s² band, as compared to analogous situations in oxides. Introducing holes into the hydride anion band by necessity means substantial weakening and even rupture of the element–hydrogen bond, in contrast to anions carrying p orbitals. This feature of H⁻ also helps to explain the low-T_C value of Th₄H₁₅. Th₄H₁₅ is formally a nonstoichiometric e⁻-doped (ThH₄)₄ (f⁰); an extra f¹ electron thus needs to sit in a band originating from Th–H nonbonding states, built predominantly from Th 5f orbitals. Thus, conducting electron density is delocalized predominantly on Th network, with no contribution of H; the metallic form of Th₄H₁₅ (a ‘low-D metal’ in Zaanen–Sawatzky–Allen diagram) cannot fully benefit from high-frequency H-based phonons. In contrast, the f¹³/f¹⁴ doping for our systems introduces changes in a (weakly antibonding) σ* Ln–H states; the vibronic effects should thus be larger than for the f⁰/f¹ perturbation.
37. Here, two important differences between hole-doping to Cu^II and to Tm^II come to mind. Hole-doping (which proved very successful in introducing superconductivity in oxocuprates), might be realized for Tm^II compounds by their partial oxidation to trivalent Tm. Note, however, that, first, f¹² Tm^III is a strongly magnetic ion for most ligand fields (magnetic moment of ∼7.5 µB), in contrast to classical low-spin d⁸ Cu^III for oxocuprate superconductors, and, second, the magnetic moment of f¹³ Tm^II itself (∼4.5 µB) is much greater than that of d⁹ Cu^II (∼1.8 µB). This means that magnetic ordering will be much easier for doped compounds of Tm than for those of Cu. Instead of introducing vacancies, other variations can also be used, e.g., Ln^IIH₂ layer could be sandwiched between Ba^IIO sheets, and then doped by substitution at Ba^II site with either K^I (hole-doping) or with Ac^III (electron-doping).
38. The electron doped BaPb_1−xBi_xO₃ superconductor does not have an isostructural lanthanide analogue. Hypothetical □₁Yb^III_1−xTm^III_xH₃ might be formally treated as such; unfortunately, parent stoichiometric YbH₃ has not been synthesized. Also La_2−xBa_xCuO₄ does not have any counterpart among the hydrides of Tm and Yb.
39. Mixed Eu^II/Yb^II and Sm^II/Yb^II but not Tm^II/Yb^II compounds have been obtained; see ref. 12.
40. D. T. Adroja and S. K. Malik, J. Magn. Magn. Mater., 1991, 100, 126–138; J. F. Herbst and J. W. Wilkins, Phys. Rev. B, 1982, 26, 1689–1701; B. Batlogg, A. Schlegel and P. Wachter, Physica B+C, 1977, 86, 229–230.
41. Valence fluctuations have been linked to superconductivity in the early days of high-T_C superconductivity; see for example: W. Y. Liang, J. Phys. C, 1987, 20, L571–L576 and have also inspired its very discovery.
42. T. Sato, D. Noréus, H. Takeshita and U. Häussermann, J. Solid State Chem., 2005, 178, 3385–3392.
43. Theoretical investigations of hypothetical LnBeH₄, containing light Be, and of ternary hydrides of trivalent lanthanides, are now in progress in our laboratory.

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Footnote

† Electronic supplementary information (ESI) available: Full DOS for CsYbH₃ and VASP results. See DOI: 10.1039/b514773e

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