# Metal Borohydrides as high-$T_{\text{c}}$ ambient pressure superconductors

Simone Di Cataldo<EMAIL_ADDRESS>Dipartimento di Fisica,
Sapienza Università di Roma, 00185 Roma, Italy  Institute of Theoretical and
Computational Physics, Graz University of Technology, NAWI Graz, 8010 Graz,
Austria  Lilia Boeri<EMAIL_ADDRESS>Dipartimento di Fisica, Sapienza
Università di Roma, 00185 Roma, Italy  Centro Ricerche Enrico Fermi, Via
Panisperna 89 A, 00184 Rome, Italy

###### Abstract

The extreme pressures required to stabilize the recently discovered
superhydrides represent a major obstacle to their practical application. In
this paper, we propose a novel route to attain high-temperature
superconductivity in hydrides at ambient pressure, by doping commercial metal
borohydrides. Using first-principles calculations based on Density Functional
Theory and Migdal-Éliashberg theory, we demonstrate that in Ca(BH4)2 a
moderate hole doping of 0.03 holes per formula unit, obtained through a
partial replacement of Ca with monovalent K, is sufficient to achieve
$T_{\text{c}}$’s as high as 110 K. The high-$T_{\text{c}}$ arises because of
the strong electron-phonon coupling between the B-H $\sigma$ molecular
orbitals and bond-stretching phonons. Using a random sampling of large
supercells to estimate the local effects of doping, we show that the required
doping can be achieved without significant disruption of the electronic
structure and at moderate energetic cost. Given the wide commercial
availability of metal borohydrides, the ideas presented here can find prompt
experimental confirmation. If successful, the synthesis of high-$T_{\text{c}}$
doped borohydrides will represent a formidable advancement towards
technological exploitation of conventional superconductors.

##  I Introduction

Since the discovery of superconductivity with a critical temperature
($T_{\text{c}}$) of 203 K at 150 GPa in H3S [1, 2], hydrogen-rich
superconductors have revolutionized the landscape of superconductivity
research; After H3S, many other superhydrides with $T_{\text{c}}$’s close to,
or even above, room temperature have been found [3, 4, 5, 6, 7, 8, 9, 10, 11,
12], but the extreme synthesis pressures represent an insurmountable obstacle
to any practical application.

On the other hand, an increasing demand exists for new materials enabling
superconductor-based technologies: for most large-scale applications,
synthesizability at ambient pressure is a strict requirement, whereas the
threshold for $T_{\text{c}}$ is sensibly lower than ambient temperature, but
is dictated by the need to maintain a robust superconducting state under
liquid nitrogen cooling. [13]

The spectacular success of computational methods for superhydrides raises the
hope that these techniques may accelerate the identification of suitable
materials.[14, 15] The first predictions which have started to appear in
literature follow essentially two different routes to boost $T_{\text{c}}$
within the conventional ($ep$) scenario: 1) optimization of the effective
chemical pressure in ternary hydrides [16, 17, 18, 19, 19, 20], through a
careful combination of guest elements in host-guest metallic superhydrides;
and 2) doping of non-hydride covalent structures [21, 22, 23, 24, 25, 26, 27],
which exploits the large intrinsic $ep$ coupling of covalent insulators,
turned metallic via either external or self doping, as in B-doped diamond or
MgB2 [28, 29, 21, 30].

Both approaches present potential drawbacks: while even the most optimized
ternary hydrides seem to require synthesis pressure of at least a few GPa
[17], the $T_{\text{c}}$’s of systems containing exclusively boron, carbon and
other heavier elements are unlikely to exceed the 80 K threshold found in
best-case scenarios [26, 27], due to the intrinsic phonon energy scales
determined by the relatively large boron and carbon atomic masses.

In this paper we propose a hybrid strategy which combines the best of the two:
doping covalent bonds (ambient pressure) in a hydrogen-rich structure (higher
$T_{\text{c}}$). In particular, we show that metal borohydrides (MBH) can be
turned into high-temperature conventional superconductors at ambient pressure,
via small substitutional doping at the metal site, which effectively
transforms MBH into highly-tunable hole-doped hydrocarbons.

MBH form a broad class of materials widely used in commercial hydrogen storage
applications, due to the high hydrogen content, and the ease of hydrogen
uptake and dehydrogenation [31, 32]. In these compounds boron and hydrogen
form quasi-molecular units arranged on open structures, with mono-, di- or
trivalent metals ($M$) on the interstitial sites. Our strategy to turn MBH
into high-$T_{\text{c}}$ superconductors is quite general, and consists in
replacing a small fraction of $M$ atoms with a lower-valence atom, realising
hole doping; in this work, we study the specific case of K- doping of the
$\alpha$ phase of Ca(BH4)2 [33].

Our calculations demonstrate that substitutional K doping in Ca(BH4)2 is
energetically feasible up to at least 0.10 h+/f.u.; concentrations as low as
0.03 holes per formula unit (h+/fu) are sufficient to induce superconductivity
with a $T_{\text{c}}$ as high as 110 K. As MBH are commercially available
materials, we expect our work to find an immediate response from experimental
researchers.

Figure 1: Crystal structure of $\alpha$-Ca(BH4)2. The Ca, B, and H atoms are
shown as green, orange, and blue spheres, respectively. BH${}_{4}^{-}$ anions
are shown as tetrahedra.

$\alpha$-Ca(BH4)2 (Fig. 1) is a molecular crystal, in which boron and hydrogen
form BH4 tetrahedra, and Ca occupies interstitial sites. Ca is almost
completely ionized (Ca++), and donates charge to the BH${}_{4}^{-}$
tetrahedra, which are thus not only isostructural, but also isoelectronic to
methane (C$H_{4}$). The spacing between BH${}_{4}^{-}$ molecular units is
quite large, about 3.5 $\AA$, indicating extremely weak intermolecular
interactions.

Fig. 2 shows the electronic band structure and the atom-projected Density of
States (DOS) of the $\alpha$ phase of Ca(BH4)2. Undoped $\alpha$-Ca(BH4)2 is
an insulator, with a calculated direct band gap of 5 eV. The bands have a
reduced dispersion, as typical of molecular crystals; the electronic DOS
exhibits extremely sharp and narrow peaks, particularly near the valence band
maximum (VBM). Electronic states in this region have a mixed (50/50) B/H
character and derive from the three-fold degenerate $1t_{2}$ (0 to -2 eV) and
the single $2a_{1}$ (-6 to -8 eV) molecular $\sigma$ orbitals of BH4, which
are expected to couple strongly to B-H bond-stretching and bond-bending,
phonons.

Due to the extremely sharp profile of the DOS, even extremely small hole
dopings are sufficient to shift the Fermi energy into the large-DOS,
large-$ep$ region below the VBM, which should induce high-$T_{\text{c}}$
conventional SC.

Figure 2: Electronic band structure and atom-projected DOS of Ca(BH4)2. The
energy zero is set to the valence band maximum. The DOS is in units of states
$eV^{-1}atom^{-1}$.

Hole doping in Ca(BH4)2 can be realized by substituting Ca with a monovalent
atom. In this work, we consider K, which is the neighbour of Ca in the
periodic table, and hence has very similar size and core. Replacing a fraction
$\delta$ of divalent Ca with monovalent K amounts to doping $\alpha$-Ca(BH4)2
with $\delta$ holes/f.u.; in an ideal rigid-band picture, these holes would
form at the top of the valence band, turning K-doped Ca(BH4)2 into a self-
doped version of methane (CH4).

Due to the presence of stiff covalent bonds coupled by symmetry to bond-
stretching phonons, doped hydrocarbons have long been postulated to exhibit
large $ep$ coupling; controversial reports of high-$T_{\text{c}}$
superconductivity in polyacenes doped with alkali and alkaline earths
(electron doping) have appeared in the early 2010’s [34, 35, 36, 37]. However,
doping hydrocarbons and related C-H systems with holes has so far proven
impossible, as intercalation with electronegative elements (I,F, Cl) is
ineffective, and doping the C-H sublattice by substitutional atoms or
vacancies is extremely unfavorable energetically and tends to seriously
disrupt the crystal and electronic structure due to the presence of stiff
covalent bonds [38, 39].

In K-doped Ca(BH4)2, on the other hand, doping only involves the metal site,
which is very weakly bonded to the rest of the structure. This should allow a
convenient fine-tuning of the superconducting properties at a reasonable
energy cost, without major modifications of the structure [40].

To substantiate our hypotheses, we computed the superconducting properties of
K-doped Ca(BH4)2 for various hole concentrations $\delta$, computing the
isotropic Éliashberg functions for various values of $\delta$ using Density
Functional Perturbation Theory [41], and obtaining the $T_{\text{c}}$ by
numerically solving the isotropic Éliashberg equations 111Calculations were
performed using DFPT as implemented in Quantum Espresso. Integration of
electron-phonon properties was performed on $3\times 3\times 3$ grid for
phonons and $16\times 16\times 16$ grid for electrons, using a Gaussian
smearing of 200 meV. Further computational details are provided in the
Supplementary Material [44, 49, 50, 41, 51, 52, 53].

Doping is simulated using the virtual crystal approximation (VCA), which
amounts to replacing each Ca (pseudo)atom in the $\alpha$-Ca(BH4)2 structure
with an average virtual (pseudo)atom, obtained by mixing K and Ca in the
appropriate proportions.

In Fig. 3 we report a summary of the electronic and superconducting properties
of K-doped Ca(BH4)2 as a function of the hole concentration $\delta$. The DOS
at the Fermi level N${}_{E_{F}}$ (panel (a)) rapidly increases with doping, as
does the total $ep$ coupling constant $\lambda$ (panel (c)), while the average
phonon frequency $\omega_{log}$ (b) slightly decreases. In the Éliashberg
function (shown in Fig. S2 of the Supplementary Material) almost all $ep$
coupling is concentrated in the high-energy B-H stretching and bending modes.
For $\delta$ larger than 0.10 the system develops a dynamical instability,
while for values smaller than 0.03 the Fermi energy is too close to the VBM to
allow a reasonable estimate of $\lambda$ and $T_{\text{c}}$. $T_{\text{c}}$
attains its maximum value of 130 K at $\delta$ = 0.10 and decreases linearly
with decreasing $\delta$ down to 110 K at $\delta$=0.03; extrapolating this
trend, we can reasonably suppose that $T_{\text{c}}$’s higher than 100 K may
be achieved for dopings $\delta\gtrsim 0.01$.

Figure 3: Electronic and superconducting properties as a function of the
doping $\delta$. (a): DOS at the Fermi level, (b): logarithmic average phonon
frequency $\omega_{log}$, (c): $ep$ coupling coefficient $\lambda$, (d):
superconducting critical temperature $T_{\text{c}}$.

The VCA has the advantage of making $T_{\text{c}}$ calculations feasible even
for small dopings, correctly capturing the average effect of substitutional
doping on the electronic structure, in particular the critical role of
electronic screening on phonon spectra and $ep$ matrix elements. [37] However,
effects such as charge localization, deep defect levels or carrier trapping
[43, 24], which may sensibly affect the electronic structure, require more
complex approximations that can capture depend on the local environment of the
impurities.

To simulate these effects, we developed an ad-hoc scheme, based on averaging
over random supercells. First, we constructed a 2$\times$2$\times$2 supercell
containing 32 formula units of Ca(BH4)2 (352 atoms); then, to simulate hole
concentrations from $\delta=0.03$ to $\delta=0.5$, we substituted Ca atoms
with the appropriate fraction of K, placed at random positions; for each value
of $\delta$, we generated ten supercells. These supercells were then relaxed
to minimize stress and forces, before computing the total energies and DOS’s –
Figs. 4. Computations on the supercells were performed using the Vienna Ab-
initio Simulation Package (VASP). Further details are provided in the
Supplementary Materials [44]. The average DOS for each doping was then
obtained by performing a weighted average over the relative supercells, with
weights corresponding to the probability of that configuration (See SM for
further details [44]). The average DOS’s for different values of $\delta$ are
shown in Fig. 4. Although doping causes sizable modifications of the DOS for
$\delta>0.12$, especially in the low-lying region below -6 eV, the DOS for
$\delta$ up to 0.09 are essentially unchanged compared to the undoped
compound. In particular, there are no inter-gap states up to $\delta$ = 0.09,
and the relative weigth remains negligible up to $\delta=0.25$.

Hence, for doping of interest the main effect of K/Ca substitution is indeed a
quasi-rigid shift of the Fermi level into the valence band, well reproduced by
VCA; in particular, the states just below the VBM, which participate in the SC
pairing, are only weakly influenced by doping. This is expected, since
$\sigma$ orbitals of the BH${}_{4}^{-}$ molecular ions are only weakly
affected by distortions and rearrangements of atoms in the crystal structure
which do not modify the overall shape of the molecular ion itself.

Figure 4: Density of states (DOS) of K-doped Ca(BH4)2 as a function of the
hole concentration $\delta$. The Fermi energy is taken as the energy zero
except for $\delta=0$, in which the VBM is used.

Using supercells we can also estimate the energetic cost of K substitution
into the Ca site. In Fig. 5 we show the formation energy of KδCa1-δ(BH4)2 with
respect to decomposition into K + Ca(BH4)2 222For Ca and K we assumed a face-
centered and a body-centered cubic structure, respectively, for all
configurations sampled (ten for each doping). The formation energies $\delta
E$ increase linearly with doping, with little dispersion for different
supercells, remaining below 200 meV/atom for $\delta$ up to 0.12.

On purely energetic grounds, these values indicate that K-doping of Ca(BH4)2
should be experimentally feasible. However, experimental synthesis conditions
depend on complex details of the kinetic barrier protecting the doped
structure from decomposition, and on the entropy contribution to the free
energy, whose evaluation goes well beyond the scope of this paper.

Moreover, independently of their energetic cost, not all perturbations induced
by doping will have the same effect on superconductivity; while small
distortions and rearrangements of BH${}_{4}^{-}$ anions within the open
$\alpha$ structure, should have only minor effects on the the superconducting
properties, dehydrogenation, which implies a weakening of the B-H bonds, has
severe consequences, since it implies a major rearrangement of the whole
electronic structure. We will therefore assume that the dehydrogenation energy
can be used to estimate an effective synthesizabilty threshold for K-doped
Ca(BH4)2.

A hand-waving estimate can be obtained as follows. Assuming the measured
dehydrogenation temperature of Ca(BH4)2, which is around 700 K [46, 47, 48],
to be a reasonable guess of the kinetic barrier for dehydrogenation, doped
Ca(BH4)2 should be able to withstand perturbations with a positive $\Delta E$
of the order of 60 meV/atom without decomposing. This corresponds to a doping
$\delta\sim$ of 0.06 (Fig. 5), which as Fig. 3 (d) shows, largely exceeds the
doping levels required for high-$T_{\text{c}}$ superconductivity. Hence,
high-$T_{\text{c}}$ superconductivity should be observable, before
dehydrogenation sets in.

Figure 5: Formation energy $\Delta E$ as a function of hole-doping $\delta$.

In summary, in this paper we proposed a strategy to attain high-$T_{\text{c}}$
conventional superconductivity in the complex bordohydride Ca(BH4)2 using
substitutional doping of monovalent K on the Ca site, and substantiated it
with first-principles calculations.

K-doped Ca(BH4)2 behaves essentially as hole-doped methane (CH4), where the
high-$T_{\text{c}}$ derives from a strong coupling between $\sigma$ bonding
electronic states and bond-stretching phonons of the BH${}_{4}^{-}$ molecular
units. Compared to CH4, however, the big advantage of Ca(BH4)2 is that hole
doping is realized by acting on the weakly-bonded metal site, and not on the
covalent B-H (or C-H) sublattice, and this causes only minor disruptions of
the crystal and electronic structure, implying an affordable energy cost.
According to our calculations, a partial replacement of 3 $\%$ Ca atoms with K
atoms would have an energy cost of around 50 meV/atom, which is below the
dehydrogenation threshold, and lead to an estimated $T_{\text{c}}$ of 110 K at
ambient pressure, almost on par with the best copper-oxide superconductors.
With a figure of merit $S$ between 2.8 and 3.3 [13], doped Ca(BH4)2 is better
than any other superhydride, as well as all other known ambient-pressure
conventional superconductors ($S=1$ in MgB2), and very close to HgBaCaCuO.

Note that the strategy proposed here is very general, and can in principle be
applied to turn any of the many existing MBH into doped hydrocarbons, by
suitable metal substitutions. We are strongly convinced that, if synthesized,
doped metal borohydrides will represent a huge leap forward in research on
high-temperature superconductors.

Given the easy commercial availability of metal borohydrides, we hope that our
work will stimulate a positive response from experimentalists.

##  II Acknowledgments

The authors warmly thank Antonio Sanna for sharing the code to solve the
isotropic Éliashberg equations. L.B. and S.d.C. acknowledge funding from the
Austrian Science Fund (FWF) P30269-N36 and support from Fondo Ateneo-Sapienza
2017-2020. S.D.C. acknowledges computational resources from CINECA, proj.
IsC90-HTS-TECH and IsC99-ACME-C, and the Vienna Scientific Cluster, proj.
71754 ”TEST”.

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