The road to room-temperature conventional superconductivity

TL;DR

This paper reviews the progress in achieving room-temperature superconductivity in conventional materials, highlighting the role of computational methods and recent breakthroughs from 40 K to 265 K.

Contribution

It discusses the development of computational techniques that enabled the rapid progress in room-temperature superconductivity over the past two decades.

Findings

Superconducting transition temperature increased from 40 K to 265 K.

Computational methods played a crucial role in discovering new superconductors.

The progress signifies a major milestone towards practical room-temperature superconductivity.

Abstract

It is a honor to write a contribution on this memorial for Sandro Massidda. For both of us, at different stages of our life, Sandro was first and foremost a friend. We both admired his humble, playful and profound approach to life and physics. In this contribution we describe the route which permitted to meet a long-standing challenge in solid state physics, i.e. room temperature superconductivity. In less than 20 years the Tc of conventional superconductors, which in the last century had been widely believed to be limited to 25 K, was raised from 40 K in MgB2 to 265 K in LaH10. This discovery was enabled by the development and application of computational methods for superconductors, a field in which Sandro Massidda played a major role.

Figures

Equations

$${\alpha }^{2}F(\omega )=\frac{1}{N(E_F)}\sum _{\mathbf{kq},\nu }|g_{\mathbf{k},\mathbf{k}+\mathbf{q},\nu }{|}^2\delta ({ϵ}_{\mathbf{k}})\delta ({ϵ}_{\mathbf{k}+\mathbf{q}})\delta (\omega -{\omega }_{\mathbf{q},\nu })\text{leavevmode},$$

$$T_c=\frac{{\omega }_{\log }}{1.2k_B}\exp \left[-\frac{1.04(1+\lambda )}{\lambda -{\mu }^{\ast }(1+0.62\lambda )}\right],\text{where}$$

$$\lambda =\frac{N(0)I^2}{M{\omega }^2},\text{where}$$

Full text

Since K.H. Onnes discovered in 1911 that a sample of mercury, cooled below a critical temperature ($T_{\text{c}}$) of 4 K, exhibits a vanishing resistivity, it became immediately clear that such a superconductivity, realized at ambient conditions, could have spectacular electrical network applications. A related unique property of superconductors, perfect diamagnetism, is equally attractive, because quantum levitation paves the road to futuristic scenarios, such as levitating vehicles.

However, more than 100 years after the original discovery, none of these large-scale applications has advanced to a point where it is economically viable. Currently, the only applications of superconductors are found in devices and facilities whose cost is not an issue: superconducting magnets are used in large-scale particle accelerators and storage rings, diagnostical devices, antennas etc. The most severe obstacle to cost-sensitive applications are the prohibitive refrigeration costs required to cool the existing materials below their critical temperatures. The only compounds which superconduct above the liquid nitrogen (N2) boiling point, the high-$T_{\text{c}}$  cuprates, have several characteristics which make them not suitable for large scale applications: being brittle, they have high manufacturing costs; the presence of magnetism in the phase diagram and the exotic symmetry of the superconducting gap make them sensitive to grain boundaries. Gurevich (2011)

As a result, within today’s technological applications, the most used superconductors are simple intermetallic alloys, with $T_{\text{c}}$’s in the range of 10-30 K. In addition to being much cheaper and easier to manipulate, these superconductors are also easier to describe theoretically: unlike the exotic cuprates, they are described to a high degree of accuracy by the theory of conventional phonon-mediated superconductivity, developed in the 60’s; the progress of current ab-initio methods to calculate the electron-phonon coupling in the last 20 years is such that the normal- and superconducting-state properties of actual materials can now be computed to a high degree of accuracy based on the sole knowledge of their chemical composition and crystal structure; actually, once the formula unit is given, the equilibrium crystal structure itself may be predicted, to a great degree of accuracy, using a combination of ab-initio structural energies and modern optimization methods, a valuable theoretical-computational tool which becomes almost indispensable when one or some of the chemical components are light elements which are hard to locate by X-ray diffraction.

The availability of quantitative methods to describe the electronic, vibrational, and structural properties of materials has granted not only an accurate understanding of many existing superconductors, but also the exploration and prediction of hypothetical, new ones, up to providing, last year, the solution to a puzzle previously deemed impossible: a room-temperature superconductor. In the summer of 2018, two different groups reported the discovery of a $T_{\text{c}}$  of 260 K in a sample of lanthanum hydride (LaH10), pressurized to Megabar pressures ($\sim 150$ GPa)  Drozdov et al. (2018); Somayazulu et al. (2019); Drozdov and et al. (2018), and already three years before another high-pressure hydride, SH3, had set the all-time superconducting record with a $T_{\text{c}}$  of 203 K Drozdov et al. (2015a); in both cases their stability and superconductive nature, and to a very good approximation their $T_{\text{c}}$, had been predicted by ab-initio calculations. Duan et al. (2014a); Peng et al. (2017)

The discovery of high-$T_{\text{c}}$  superconductivity in high-pressure hydrides does not solve any technological problem, since, in daily life, to achieve pressures million times larger than ambient pressure is much more difficult than refrigerating samples down to a few K. Nonetheless, the discovery of room-temperature superconductivity in LaH10  has smashed several psychological barriers, showing: ($i$) that superconductivity may be achieved at ambient temperature; ($ii$) that high-$T_{\text{c}}$  superconductivity may be achieved by the conventional (electron-phonon) mechanism; ($iii$) that, using first-principles calculations, new superconductors may be reliably predicted (and designed) before experiments.

Obviously, the achievement of room-temperature superconductivity was not a matter of sheer luck, but rather the result of a long process, which experienced a strong acceleration at the beginning of this century. The aim of this viewpoint is to illustrate the last steps of this process, which took place in the last 20 years. Out of the rich literature on the subject, we have identified four milestones, illustrated in Fig. 1, which marked the route to the LaH10  discovery.

The discovery of MgB2  and other covalent superconductors (2001-), red circles. 2. 2.

The study of elemental superconductors at high pressures (2002-), orange circles. 3. 3.

The theoretical calculation of the phase diagram and $T_{\text{c}}$’s of metallic Hydrogen. (2007-), green circle. 4. 4.

The prediction of superconductivity in high-pressure hydrides (2008-), blue circles.

The two last milestones are directly related to two brilliant intuitions on solid hydrogen and hydrogen-rich hydrides due to Neil Ashcroft Ashcroft (1968, 2004) which we will discuss later. It is also interesting to note that, at the start of the route, the experimental discovery of the remarkable superconducting $T_{\text{c}}$ of MgB2 preceded its understanding in terms of ab initio electronic bands and phonons, whereas, towards the end of the route, theoretical predictions based on large-scale ab initio calculations led experimental groups to synthesize previously unknown hydrides like SH3 or LaH10, eventually confirming both their stability and their exceptionally high $T_{\text{c}}$. In other words, the development and extensive application of ab-initio methods was so impetuous as to turn them, in less than 20 years, from valuable instruments for the interpretation of existing superconductors into reliable forecasting tools for new superconductors. Before entering the route leading to the discovery of LaH10, we need, however, to rapidly recall the basic theoretical background of superconductivity, a field to which Sandro, with his students and collaborators, gave an essential contribution.

The simplest microscopic theory of superconductivity, the Bardeen-Cooper-Schrieffer (BCS) theory, describes the superconducting state as a state of macroscopic quantum coherence, where electrons form pairs of opposite spin and momentum (Cooper Pairs) held together by an attractive glue. The superconducting state is characterized by the presence of a gap $\Delta$ in the superconducting spectrum, which closes at the critical temperature $T_{\text{c}}$; above $T_{\text{c}}$, conventional superconductors behave as normal metals. In the original BCS paper, the glue is provided by lattice vibrations (phonons), but other mediators are in principle possible, such as plasmons, spin fluctuations, charge fluctuations, etc.

The strong-coupling extension of BCS theory, known as Migdal-Eliashberg (ME) theory, is a diagrammatic theory for interacting electrons and bosons Allen and Mitrović (1983). At the heart of the theory is the so-called (isotropic) electron-phonon spectral function, defined as:

[TABLE]

where $N(E_{F})$ is the DOS at the Fermi level, $\omega_{\mathbf{q},\nu}$ is the phonon frequency of mode $\nu$ and wave vector $\mathbf{q}$, and $|g_{\mathbf{k},\mathbf{k}+\mathbf{q},\nu}|$ is the e-ph matrix element between two electronic states of wave vectors $\mathbf{k}$ and $\mathbf{k+q}$ at the Fermi level. The expression of the e-ph spectral function $\alpha^{2}F(\omega)$ clearly evidences why superconductors are so hard to predict (or why superconducting properties are so sentitive to small details of the electronic structure of a given material): since in Eq. (1) the phonon spectrum is weigthed by the e-ph matrix elements $|g_{\mathbf{k},\mathbf{k}+\mathbf{q},\nu}|$, which can be very different for different phonon modes, and since the double delta function $\delta(\varepsilon_{\mathbf{k}}^{n})\delta(\varepsilon_{\mathbf{k+q}}^{m})$ restricts the sum of $e\!-\!ph$ matrix elements to electronic states at the Fermi level, only a small fraction of the full electronic and phononic spectrum contributes to superconductivity.

To understand and predict new conventional superconductors essentially amounts to explaining or conceiving materials where electrons and phonons are strongly coupled (large $|g_{\mathbf{k},\mathbf{k}+\mathbf{q},\nu}|$), which can be done with considerable accuracy since all quantities entering the e-ph spectral function can be computed from first-principles within Density Functional Perturbation Theory Baroni et al. (2001). Once the e-ph spectral function is known, the critical temperature of a material can be computed to different degrees of sophistication. The simplest approach adopts an approximate formula for $T_{\text{c}}$; a popular choice for phonon-mediated superconductors is the Mc-Millan-Allen-Dynes formula Allen and Dynes (1975)

[TABLE]

$\lambda\!=\!2\int d\omega\frac{\alpha^{2}F(\omega)}{\omega}$ and $\omega_{\log}\!=\!\exp\left[\frac{2}{\lambda}\int\frac{d\omega}{\omega}\alpha^{2}F(\omega)\ln(\omega)\right]$ are the e-ph coupling constant and logarithmic-averaged phonon frequency; $\mu^{*}$ is the so-called Morel-Anderson pseudopotential, obtained by screening the full Coulomb potential up to a characteristic cut-off energy. Morel and Anderson (1962)

This approach neglects effects which may influence the behavior of a superconductor, such as possible anisotropies of the superconducting gap, strong-coupling corrections to the $T_{\text{c}}$  expression, self-energy effects on the electronic and phonon spectra, etc. These effects may be handled from first principles in two ways: ($i$) on one hand, the so-called anisotropic Migdal-Eliashberg Theory (AMET), which solves the anisotropic Migdal-Eliashberg equations, computing the full electronic and phonon self-energies; although very accurate, it cannot be considered a fully ab-initio theory, but rather an advanced combination of many-body techniques and first-principles ingredients; ($ii$) on the other hand, the superconducting Density Functional Theory (SCDFT) proposed by Oliveira, Gross and Kohn in 1988 Oliveira et al. (1988) but implemented for real materials only at the beginning of the 2000’s by Sandro Massidda and Hardy Gross’ groups, a conceptually different extension of the DFT approach to the the superconducting state Lüders et al. (2005); Marques et al. (2005).

Both AMET and SCDFT remove two strong assumptions of the simpler ME version, i.e. the isotropy of the e-ph coupling and superconducting gap and the use of an empirical Morel-Anderson pseudopotential. Excellent reviews of the latest developments of the two approaches may be found in Refs. Giustino, 2017-Sanna et al., 2018, and Sanna, 2017, respectively. For our historical perspective, it is sufficient to say that the two approaches have by now reached comparable accuracy ($\sim 10\%$ on the $T_{\text{c}}$), and that a formal mapping of the two theories is possible using the Sham-Schlüter connection Sham and Schlüter (1983).

The first attempt to calculate the $T_{\text{c}}$ of a conventional superconductor from first principles dates back to the 1970’s, when Gaspary and Gyorffy proposed an approximate method to compute the e-ph coupling in superconductors Gaspari and Gyorffy (1972). The deformation-potential approach by Kahn and Allen is ten years older Khan and Allen (1984). Both approaches were too crude to yield reliable results; a serious limitation was represented by the prohibitive cost of computing phonon spectra in those years: they were fitted to experiments via semi-empirical force-constant models.

A big progress came with the development of Density Functional Perturbation Theory (DFPT) Baroni et al. (2001). On this basis, Savrasov and Savrasov published in 1996 the first complete set of $T_{\text{c}}$  calculations, showing that DFPT, combined with a semi-phenomenological ME approach, was capable of a reliable description of the superconducting and transport properties for all known elemental superconductors at ambient pressure, and also of the explanation why some elements, like Cu or Pd, are not superconductors Savrasov and Savrasov (1996). The importance of this work was recognized only a few years later, because, at that time, the superconductivity debate was dominated by the cuprates, where such a conventional e-ph mechanism for the high-$T_{\text{c}}$  had been clearly ruled out Savrasov and Andersen (1996).

Moreover, an old semiempirical argument based on the simplest version of the ME theory predicted an upper limit of 25 K for the $T_{\text{c}}$  of conventional superconductors –the so-called Cohen-Anderson limit– and this seemed to be confirmed by the experimental discoveries: year by year $T_{\text{c}}$  had been slowly and painfully pushed up to 25 K in the best superconductors known at that time, the A15, (Nb3Ge), and it seemed impossible to go beyond that value; cuprates were not an exception to this rule, since their superconductivity was soon recognized as due to a coupling mechanism other than e-ph coupling.

This is why, in our recollection of the path to room-temperature superconductivity, a much less spectacular discovery paradoxically appears more important than that of cuprates: in 2001, a simple intermetallic, magnesium diboride (MgB2), was found to be superconducting with a $T_{\text{c}}$  of 39 K. Nagamatsu et al. (2001) Besides disproving the Cohen-Anderson limit for $T_{\text{c}}$, MgB2  has characteristics which make it stand out of the group of previously known conventional superconductors: its superconducting gap has very different values on different sheets of its Fermi surface (two-gap superconductivity) and its phonon spectrum is strongly anharmonic.Canfield and Crabtree (2003) Fortunately the description of such effects requires a relatively straightforward extension of the standard theory of conventional superconductivity which can be incorporated into ab-initio approaches: Kong et al. (2001); Kortus et al. (2001); Liu et al. (2001); Choi Hyoung Joon et al. (2002) the MgB2 case induced important methodological developments, such as the implementation of the anisotropic formalism for e-ph interaction, the development of Wannier-function methods to accurately compute linewidths and Kohn anomalies, a new formalism to compute self-consistently anharmonic effects on phonon spectra, etc.  Giustino et al. (2007); Calandra et al. (2010); Errea et al. (2014) But the most important role played by the MgB2  discovery was probably to provide a new playground for the understanding of material-specific mechanisms which within the conventional e-ph coupling may give rise to a high $T_{\text{c}}$.

Until 2001, the best conventional superconductors were intermetallics containing $d$ metals, and the main strategy adopted to “boost” their $T_{\text{c}}$  was the use of different dopants to increase their density of states at the Fermi level. Unlike them, MgB2 was a simple $s$–$p$ material whose high-$T_{\text{c}}$  derived from “covalent bonds driven metallic”, as nicely synthesized by the title of a paper by Warren Pickett An and Pickett (2001).

The meaning of these expressions is most easily understood if one rewrites the e-ph coupling constant $\lambda$ using the simplified Hopfield expression:

[TABLE]

$N(0)$ is the density of states at the Fermi level, $I$ the e-ph matrix element, and $M\omega^{2}$ a phonon force constant.

In most elemental superconductors and, to a lesser extent, in A15 compounds, the e-ph coupling spreads over several phonon modes and electronic states; the highest $T_{\text{c}}$’s are obtained from the largest $N(0)$ values (typically within narrow transition-metal bands) combined with a moderate average e-ph  matrix element and relatively low phonon frequencies, and remain limited to $\sim$ 25 K.

On the other hand, MgB2 (and also, as we shall later see, some high-pressure hydrides) have relatively small $N(0)$’s. What drives the high $T_{\text{c}}$, in these cases, are large e-ph matrix elements ($I$) between a few selected bond-stretching phonons and the electrons which contribute to those directional, covalent bonds. However, such electrons are shared by neighboring atoms within fully occupied bonding states which are well separated in energy from the corresponding, empty antibonding states; as a result, under normal external pressure, almost all covalent or covalent-polar solids are insulators with $N(0)\!=\!0$. The few known exceptions to the thumb rule “covalent materials are insulators” all turned out to be superconductors, with higher or lower $T_{\text{c}}$ ’s primarily depending on how large or small is $N(0)$: MgB2  ($\sim$40 K), B-doped insulators ($\sim$10 K) Boeri et al. (2004); Lee and Pickett (2004), and also hypothetical compounds like hole-doped LiBC or graphane Rosner et al. (2002); Savini et al. (2010) for which $T_{\text{c}}$ ’s of $\sim$100 K were theoretically predicted. Briefly, the experimental discovery of MgB2  induced a fresh-mind theoretical re-examination of the balance among different material-specific ingredients of a high-$T_{\text{c}}$  conventional superconductor, which in turn ignited a big hunt for covalent and light-element superconducting compounds. In a few years, several classes of new (conventional) superconductors were experimentally discovered and theoretically interpreted with ab-initio calculations Ekimov et al. (2004); Blase et al. (2009); Weller Thomas E. et al. (2005).

The above discoveries are very important, but not directly related to the recent achievement of room-temperature superconductivity, so here we will not discuss them any further. We will, instead, shift to a topic which is at first sight unrelated to superconductivity.

In the same years as the MgB2  discovery, major progresses took place in the field of high pressure research McMillan (2002). Better diamond anvil cells (DAC) equipped with innovative experimental setups allowed to perform in-situ resistivity, susceptibility, Raman and IR-spectra measurements up to pressures in the Megabar range. As a result, in the first years of the 21st century the phase diagram of many compounds was explored to an unprecedented degree of accuracy. As more and more information on the behavior of matter under extreme pressure conditions piled up, it became clearer and clearer that, under pressure, even the structural evolution of the simplest elemental solids defies the chemical understanding based on ambient-pressure experience. Widely accepted concepts, such as the idea that, under sufficiently high pressure, all elements eventually form close-packed structures, urgently needed to be revised.

A 2005 survey of the superconducting properties of all elemental solids shows that, under appropriate conditions, almost all elements of the Periodic Table can be made superconducting, including those which are insulating at ambient pressure, like silicon, boron or oxygen Buzea and Robbie (2005).

Sandro’s group has played an important role in high-pressure superconducting research since its early days, when it appeared a mere intellectual curiosity. High-pressure experiments looked like an ideal testbed for the newly-developed SCDFT method. Particularly instructive, in this respect, are his early works on lithium and other alkali metals under pressure, which discuss in detail the role of phonon softening and demonstrate an anomalously strong role of the residual electronic screening in suppressing the $T_{\text{c}}$ Profeta et al. (2006). Sandro’s interest in high pressure remained high; his most recent work on superconductivity in sulfur appeared in 2017 Monni et al. (2017).

A development which was crucial to interpret high-pressure superconductivity experiments was ab-initio crystal structure prediction (CSP), which was also developed in the same years M. and Richard (2008). The basic idea of CSP is to use efficient search and optimization methods to find the global minimum of a complex energy landscape, representing a given compound at given external conditions. If the search space is sufficiently large, these methods often permit to identify the actual ground-state structure of a system. Such a possibility to predict the crystal structure of a material from first principles is particularly attractive when diffraction experiments are difficult or impossible to perform. Two of the early successes of ab-initio crystal structure prediction were the identification of the high-pressure superconducting phases of boron (max $T_{\text{c}}$  11 K) Eremets et al. (2001) and calcium ($T_{\text{c}}$  25 K)  Yabuuchi et al. (2006), two light elements with low X-Ray cross sections. After their equilibrium crystal structures were determined by CSP methods, ab initio calculations reproduced their experimental $T_{\text{c}}$’s with an accuracy of a few K, as well as their experimental pressure behavior Ma et al. (2004); Yao et al. (2009).

In the various “Periodic Tables of Superconducting Elements” published over the last ten years the first element, hydrogen, is always missing. The reason is practical: the pressure required to metallize hydrogen is at the limit of todays’ capabilities.

But as soon as the (huge) metallization pressure is reached, metallic hydrogen is expected to be a superconductor, and a good one, with very high $T_{\text{c}}$ . The original argument, proposed by Neil Ashcroft as early as 1968 Ashcroft (1968), can be understood on the basis of Eq. 2: ($i$) Due to its low atomic mass, the characteristic frequencies of hydrogen are high (prefactor $\omega_{\log}$); ($ii$) Due to the lack of screening from core electrons, electron-ion matrix elements are high (exponent $\lambda$); this means that, even with a moderate $N(0)$, $T_{\text{c}}$’s can be quite high. Ashcroft’s argument is so general, that it does not require any specific assumption on the crystal structure of hydrogen. Based on the simplified understanding of elemental crystals under pressure, early ab-initio calculations of the superconductivity of metallic hydrogen typically assumed fcc structures, and predicted $T_{\text{c}}$’s as high as 600 K Min et al. (1984); Maksimov and Savrasov (2001).

The actual high-pressure phase diagram of hydrogen turned out to be, however, much more complex than initially thought. A very important step in its understanding is a computational study of the Cambridge group, published in 2007 and based on ab-initio random structure searches, identified several high-pressure phases for which only limited spectroscopic information was, at the time, available Pickard and Needs (2007).

At ambient pressure, hydrogen forms a molecular crystal in which the H2 molecules are loosely arranged on a disordered lattice; in this structure, hydrogen exhibits a gap as large as 10 eV. Under increasing pressure, the H2 molecules tend to arrange on more and more regular lattices, giving rise to a sequence of phase transitions. In one of these molecular structures, a band-overlap insulator-to-metal transition is predicted to occur at $\sim 400$ Gpa. Slightly after (at $\sim 500$ GPa), another transition is predicted to occur towards an atomic ($\beta-Sn$) structure in which the molecular units are dismantled; this structure is also metallic.

In 2008, Sandro’s group published what can be considered the first ab-initio calculation of the $T_{\text{c}}$  of metallic hydrogen in a physically-meaningful structure Cudazzo et al. (2008, 2010a, 2010b). The calculation assumes a $Cmca$ structure, a simplified version of the intermediate high-pressure molecular structures in which one can still identify two different inter and intra-molecular bonds. As the length of these two types of bonds becomes comparable, the gap between the bonding and antibonding states closes, and hydrogen undergoes an insulator-to-metal transition. The resulting Fermi surface comprises both holes (bonding) and electron (antibonding) pockets. According to Sandro and collaborators, this should lead to a complex three-band structure of the superconducting gap. The corresponding predicted $T_{\text{c}}$’s (up to 300 K at 500 GPa) were one order of magnitude higher than what had been observed in other elements, confirming on a quantitative basis Ashcroft’s 1968 intuition that, at least in principle, high-$T_{\text{c}}$  conventional superconductivity was not impossible. More recent calculations for the atomic $\beta-Sn$ phase predict an equally large $T_{\text{c}}$’s. McMahon and Ceperley (2011); Borinaga et al. (2016)

Until a few years ago the metallization of hydrogen was beyond reach, but at least three experimental groups have reported it in the last two years, at pressures ranging from 360 to 500 GPa (3.6 to 5 Mbar)Dalladay-Simpson (2016); Eremets et al. ; Dias and Silvera (2017); the crystal structure of the metallic phase and hence the mechanism of metallization remain, however, controversial. Direct evidence is missing and computed transition pressures have a large uncertainity due to quantum lattice effects, which in hydrogen may strongly affect the relative stability of different phases as well as the phonon spectra McMahon et al. (2012); I. Errea and M. Calandra and C. J. Pickard and J. R. Nelson and R. J. Needs and Y. Li and H. Liu and Y. Zhang and Y. Ma and F. Mauri (2016). Aside from its experimental realization, which looks closer and closer, the main relevance of superconducting hydrogen is not practical, because of the huge metallization pressure required, but rather theoretical, as the starting idea whose developments eventually led to the first actual discovery of conventional high-$T_{\text{c}}$  superconductivity.

At the beginning of the century covalent hydrides were intensely studied for their ability to incorporate and release hydrogen under appropriate conditions of temperature and pressure Yang et al. (2010). Efficient hydrogen-storage materials are at the heart of hydrogen fuel cells, a clean alternative to fossil fuels for on-board automotive applications. They received a strong attention in a period when, for political and technological reasons, it seemed highly likely that the growing demand for fossil fuels would be hard to meet in the near future – See Ref.  URL, .

In 2004, Ashcroft conjectured that the ability of these hydrides to trap a large fraction of hydrogen in a host lattice could be exploited to exert a chemical pressure on hydrogen, thus lowering its metallization pressure Ashcroft (2004). The first experiments on silane (SiH4) were perfomed in 2008 and confirmed Ashcroft’s intuition, but $T_{\text{c}}$’s were disappointingly low Eremets et al. (2008). Such a low $T_{\text{c}}$  did not imply that the strategy to metallize hydrogen using hydrides was wrong per se. In hydrides, just like in other classes of superconductors, small details of the electronic structure which depend on crystal structure, chemical composition, doping, and other intrinsic or external conditions, would probably have a crucial impact on $T_{\text{c}}$ . Superconductivity was indeed close, but the chemical composition yielding a high-$T_{\text{c}}$  hydride was not as trivial as originally hoped.

At this point, the experience gained with elemental solids at high pressure turned out to be extremely precious: it had been convincingly demonstrated that, in fact, ab-initio methods for crystal structure prediction, combined with methods to compute the critical temperatures of superconductors, could be used to calculate accurate phase diagrams. Hydrides, being binary systems, pose the additional complication that several compositions may form and cohexist as a function of pressure; however, estimating the relative stability of different compositions by first-principles calculations through the convex hull constructution is rather straightforward van de Walle and Ceder (2002).

In the years immediately following Ashcroft’s prediction, many binary hydrides were computationally investigated, in hope to identify prospective high-$T_{\text{c}}$  superconductors. A very important milestone in this process is a paper by Zurek et al., published in 2009, where, for the first time, it was explicitly pointed out that off-stoichiometry phases of hydrides, in particular the superhydrides which form at high pressure, exhibit very different properties from their ambient-pressure counterparts, and some of them also show high-$T_{\text{c}}$  superconductivity Zurek et al. (2009).

The real breakthrough came in 2015, when, for the first time, a $T_{\text{c}}$  exceeding 200 K was measured in pressurized SH3, which forms when sulfur di-hydride (SH2) is placed in a hydrogen-rich atmosphere and compressed in a DAC above $\sim 10$ GPa Drozdov et al. (2015b). The H2S molecule, which is analogous to water, at ambient pressure forms a molecular, insulating crystal; in the low-pressure phases of SH3, also insulating, hydrogen is trapped in molecular form (H2) in the open H2S lattice. As pressure is increased above 20 GPa, hydrogen is gradually incorporated into the SH2 lattice, and, at $\sim$ 100 GPa, SH3  units start to form. This causes a gradual insulator-to-metal transition. At 150 GPa the S-H interaction becomes so strong that SH3  orders in a highly-symmetric ($bcc$) structure with a rather unusual bonding: sulfur forms covalent bonds with its three hydrogen neighbours. In analogy to MgB2 , these bonds, driven metallic, are at the origin of the high-$T_{\text{c}}$  conventional superconductivity of SH3 Bernstein et al. (2015); Heil and Boeri (2015).

The discovery of high-$T_{\text{c}}$  superconductivity in SH3  is an impressive demonstration of the progress of current high-pressure experimental techniques, but it also represents the first unquestionable breakthorugh of the ab-initio approach to the search and discovery of new superconductors: it was, in fact, a theoretical DFT paper Duan et al. (2014b) which, a few months before the experimental report, predicted the exact pressure, structure, $T_{\text{c}}$  and chemical composition of superconducting SH3.

This was not the only succesful prediction of high-$T_{\text{c}}$  superconductivity in high-pressure hydrides. A few months later, several theoretical papers pointed out that the neighbour of sulfur in the Periodic Table, phosphorus, should also form superconducting hydrides at high pressures, with $T_{\text{c}}$’s of $\sim 100$ K, but also that all these superconducting phases would be metastable with respect to elemental decomposition Fu et al. (2016); Eremets ; Flores-Livas et al. (2016). The metastability of such samples was later confirmed experimentally Drozdov et al. (2015c); Eremets .

Even more impressive is the case of lanthanum: in 2017 a computational study predicted that it forms superconducting superhydrides with $T_{\text{c}}$’s as large as 300$\leavevmode\nobreak\ K$ Peng et al. (2017), and in the summer of 2018 two different groups reported that one of these superhydrides, LaH10, does indeed form in a DAC, and their highest reported $T_{\text{c}}$  (265 K) is close to room temperature.

LaH10  belongs to a larger class of sodalite-like clathrates of chemical formula $X$Hn which, according to theoretical predictions, should also include the hydrides of yttrium, scandium, magnesium, calcium and several rare earths Wang et al. (2012); Peng et al. (2017). In these compounds hydrogen forms interconnected polyhedral cages each enclosing a guest atom $X$, yielding a dense sponge-like hydrogen lattice in which the shortest H-H distances are close to those predicted for metallic atomic hydrogen in the $\beta$-Sn structure. In fcc LaH10 , for example, each lanthanum sits in the middle of a 32-hydrogen cage; in bcc LaH6, the lanthanum’s nearest neighbors are 24 hydrogen atoms; and so on. Many such superhydrides with a sodalite-like clathrate structure have been predicted to exhibit high-$T_{\text{c}}$  conventional superconductivity, the highest $T_{\text{c}}$’s being expected for yttrium, calcium and magnesium Bi et al. (2018).

After the spectacular demonstration of their predictive power not only for the new high-$T_{\text{c}}$  compound SH3 , but also for the possibility of room-temperature conventional superconductivity in LaH10 , a natural question to ask is: what is the next goal of electronic-structure methods?

In our view, the first aim is to understand why some hydrides achieve room-temperature superconductivity, while others don’t. This is a step beyond the state-of-the-art ability of computing and successfully predicting phase diagrams and superconductive $T_{\text{c}}$’s of different hydrides from first principles, and amounts to identifying the specific mechanisms leading to high-$T_{\text{c}}$  in different hydrogen compounds.

First of all, one has to acknowledge the evidence of qualitatively different families within the high-pressure hydrides. For example, the high-$T_{\text{c}}$  superconductivity mechanism cannot be the same in SH3  and LaH10 : at variance with SH3 , where sulfur is incorporated into the H sublattice, forming covalent S-H bonds, the main role of the guest atom in sodalite-like clathrates is to provide charge and stabilize the the H sublattice by “filling the holes” in the sponge-like geometry.

Then one has to recognize that H-H distances close to those realized in atomic $\beta$-Sn hydrogen are a necessary, but not sufficient condition for high-$T_{\text{c}}$  superconductivity. For example, when first synthesized, the crystal structure of FeH5 (which can be seen as a cubic FeH3 lattice intercalated by two-dimensional H layers), seemed a promising candidate for high-$T_{\text{c}}$  superconductivity Pépin et al. (2017), because of the propitious H-H distances in the intermediate layer; first-principles calculations demonstrated that, instead, the Fe-H bonds which dominate the electronic states at the Fermi level are too weak to support it Christoph Heil (2018).

The second aim, directly connected to the first, is to understand whether and how superconductivity in high-pressure hydrides can have any impact on actual applications. This obviously means to seek and achieve a substantial reduction of the metallization and superconducting pressure, a goal which may be sought according to different strategies:

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already known hydrides may be used a starting point for optimization strategies. For example, in sodalite-like clathrates, the evident correlation between the size and valence of the guest atom and the stabilization pressure may be systematically studied and exploited to lower the effective metallization pressure Heil et al. ;

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ternary hydrides may offer an even larger flexibility to stabilize different crystalline superhydride phases Kokail et al. (2017), or to achieve metallization at lower pressure via doping of molecular phases Flores-Livas et al. ; Flores-Livas, José A. et al. (2018);

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compounds containing light elements other than hydrogen, although their high-pressure behavior can be quite different from hydrides, Naumov et al. (2015) may also be made superconducting under high pressure Kokail et al. (2016); among them the elements forming covalent bonds at ambient pressure, like boron and carbon, are particularly promising, since, when doped, these bonds have already proved to support high $T_{\text{c}}$. As a matter of fact, the current record for conventional superconductivity at ambient pressure (56$\leavevmode\nobreak\ K$) has been achieved last year in amorphous boron-doped carbon Bhaumik et al. (2017), and even higher $T_{\text{c}}$’s have been predicted in (ambient-pressure) doped hydrogenated graphene (graphane) Savini et al. (2010).

In summary, first-principles calculations of the superconducting properties of conventional superconductors, a field of research pioneered by Sandro Massidda, have by now gained the status of a reliable and efficient tool not only to interpret existing superconductors, but also to predict new ones, and are likely to play a decisive role in the strategies aimed at achieving ambient-pressure high-$T_{\text{c}}$  superconductors.