The 2021 room-temperature superconductivity roadmap

Designing materials with advanced functionalities is the main focus of contemporary solid-state physics and chemistry. Research efforts worldwide are funneled into a few high-end goals, one of the oldest, and most fascinating of which is the search for an ambient temperature superconductor (A-SC). The reason is clear: superconductivity at ambient conditions implies being able to handle, measure and access a single, coherent, macroscopic quantum mechanical state without the limitations associated with cryogenics and pressurization. This would not only open exciting avenues for fundamental research, but also pave the road for a wide range of technological applications, affecting strategic areas such as energy conservation and climate change. In this roadmap we have collected contributions from many of the main actors working on superconductivity, and asked them to share their personal viewpoint on the field. The hope is that this article will serve not only as an instantaneous picture of the status of research, but also as a true roadmap defining the main long-term theoretical and experimental challenges that lie ahead. Interestingly, although the current research in superconductor design is dominated by conventional (phonon-mediated) superconductors, there seems to be a widespread consensus that achieving A-SC may require different pairing mechanisms. In memoriam, to Neil Ashcroft, who inspired us all.

Machine learning has the potential to accelerate the discovery of novel superconductors, like H 3S, by reducing the computational cost of obtaining Tc from first-principles calculations and narrowing down the list of candidate materials.Using data from the Supercon database, Stanev et al. carried out machine learning studies to establish connections between the superconducting transition temperatures and atomic properties.They discovered several properties that correlate with the superconducting transition temperatures and predicted a host of compounds as possible hightemperature superconductors [6].Hutcheon et al. constructed neural network models to predict superconductors with high Tc under lower pressure than required for superhydrides thus far [7].Discovering novel materials by machine learning methods, both with and without coupling to firstprinciples calculations, is progressing rapidly.
For a given material's system, once thermodynamically stable and metastable crystal structures at various conditions are known, their physical properties, including superconductivity, can be estimated and used to guide the experimental investigations of these materials.Many techniques and codes are available for predicting potentially stable phases and structural transitions, such as the semi-local methods of minima hopping, basin hopping, and simulated annealing and the global methods of genetic algorithms and particle swarm optimization.Chemical intuition, structurechemistry correlations, and data mining of open-access materials databases can also be used to provide starting structures to these methods.In our work, we have utilized the Genetic Algorithm for Structure and Phase Prediction (GASP) package [8] for exploring possible structural transitions to clathrate-like structures at high pressures in materials containing light elements.While searching for potential stable crystal structures at high pressures, it is essential to also consider the stability of the competing phases that might destabilize the stoichiometry of the material of interest.To this end, the GASP package can perform grand canonical searches, meaning it can search for stable crystal structures over the entire composition range of interest and generate a high-pressure convex hull diagram.
While the discovery of the high-pressure superconducting hydrides has revolutionized the field of electron-phonon superconductivity scientifically, it is of little practical utility unless methods can be found to engineer materials at high pressure that remain metastable under ambient conditions.One approach has been to search for hydrides that are superconductors at lower pressures [9].However, the route to ambient pressure high-T c materials along this path is far from clear.One promising approach to ambient pressure superconductors that may be used to make large scale films is non-equilibrium growth of metastable materials on solid substrates by methods such as molecular beam epitaxy or chemical vapor deposition, potentially assisted by laser heating.Employing highenergy pre-mixed amorphous starting materials to reach structures and compositions that are inaccessible by just compressing ambient-pressure crystal structures could provide synthesis routes to novel high-pressure phases that remain metastable at ambient conditions.Theory can assist by predicting and characterizing metastable crystal structures to guide the choice of atomic constituents and estimate barriers to the equilibrium phase.

Current and Future Challenges
Machine learning and high-throughput methods work best when applied to large datasets.Given the improvement of first-principles based predictions of Tc for electron-phonon based superconductors in recent years, one could imagine constructing a database to enable machine learning of higher-T c superconductors by calculating α 2 F and associated moments, e.g., for all 16,414 materials in the SuperCon database.However, such a program is prohibitively expensive since each α 2 F calculation typically requires O(100-1000) core-hours of computing time even for elemental superconductors with a modern package such as Quantum Espresso.Purely electronic descriptors like the Fermi level density of states, N(0), are much cheaper and can enable extensions of machine learning approaches such as that attempted by Stanev et al. [6].The challenge, in this case, is to identify which purely electronic descriptors are important for high-Tc.We are constructing a high-throughput scheme along these lines.
Any high-throughput approach for novel superconductors requires robust structure predictions.Genetic algorithm searches and other methods coupled to first-principles relaxations are often limited in their prediction capabilities by the number of atoms in the supercell.As the number of atoms increases, the number of local minima in the energy landscape rises exponentially and the computational cost of relaxations grows polynomially.The room-temperature superconductors of tomorrow might potentially have large unit cells and may contain more than 3 elements.The crystal structure prediction of such superconductors can be accelerated by utilizing machine-learned surrogate models of the energy landscape that are trained on small structures.
Many of the techniques used until now in search of high-temperature electron-phonon superconducting materials may be applied to the analysis of unconventional superconductors, where Cooper pairs form due to the exchange of electronic rather than phononic excitations.Examples of unconventional superconducting materials are cuprates, iron-based superconductors, and heavy fermions.Unlike electron-phonon systems, however, a unified theory of superconductivity that could explain the variety of phenomenology shown by the various families of unconventional superconductors is not yet available.Approximate, often successful but uncontrolled methods formulated based, for example, on spin fluctuation-mediated superconductivity [9] have not been convincingly coupled to first-principles electronic structure calculations, and therefore DFT-based descriptors important for unconventional superconductivity are less well understood.Recently, we extended the approach to study a much larger sample of α 2 F's generated from a) EPW electron-phonon calculations for 50 real materials and 5,000 artificial α 2 F's of bimodal form.The solution of the isotropic Eliashberg equation, given α 2 F, is relatively inexpensive.Using the resulting dataset, which is two orders of magnitude larger than that of the previous work [3], we obtained a predictive model with improved performance over the AD formula for systems with higher T c. Figure 1a illustrates the performance of the AD formula on low-Tc materials such as elemental metals, hydrides such as LaH 10 and H3S, and the bimodal einsteins used to augment the training dataset.The systematic underprediction of TC with the AD formula is absent from predictions made with a new machine-learned equation, shown in Figure 1b.Moreover, we were able to determine that deviations of AD from Eliashberg theory, for large λ systems, arose primarily from the inequivalence of the α 2 F moments ω2 and ωlog.

Advances in Science and Technology to Meet Challenges
Based on the success of augmenting the data using model systems based on multimodal einsteins, it is clear that generative models are another promising avenue for improved machinelearning of T C. Figure 1c and 1d illustrate one method of generating α 2 F spectra using an autoencoder neural network.Training data, including known examples of α 2 F, are first discretized through binning or basis set expansion.The autoencoder learns an efficient compression of the data into a lowerdimensional "latent" space, shown here as dimensions x 1 and x2, by iteratively minimizing the error between the input and the reconstructed output.The trained decoder, comprised of the layers after and including the autoencoder bottleneck, can then transform any sampled point in the latent space into a α 2 F. Coupled with the Eliashberg equations, this approach augments the dataset with arbitrarilymany training samples that are qualitatively realistic compared to model systems.
For unconventional superconductors, a more promising approach is to use empirical knowledge about the normal state of systems that support unconventional superconductivity.For example, unconventional superconductivity is often found near phase transitions where magnetism disappears via doping or applied pressure.Such "soft" magnetic states can be identified by highthroughput DFT calculations of magnetic candidate materials (mostly metallic compounds in materials database with transition metal ions), calculating how magnetism responds to applied pressure.Promising materials where magnetization decreases rapidly can then be synthesized and studied further by doping and/or pressure.A more theoretically guided approach is based on a weak-coupling scenario in which spin-excitations provide the effective attractive interactions between electrons.In this case, our strategy is to look for the optimal conditions to find a large spin susceptibility peaked at a particular momentum that could reflect in a strong pairing at this wave vector.Nesting wave vectors on Fermi surfaces are often proxies for such peaks and can also be searched for in high-throughput schemes.

Concluding Remarks
We have outlined an approach to superconductor discovery that seeks to identify the electronic properties of a material that are essential for high T c via machine learning techniques.One thrust is to develop improved equations for T c of the Allen-Dynes type capable of accounting for large-λ materials like high-pressure superconducting hydrides.We discussed the challenges of creating sufficiently large databases of Eliashberg α 2 F's using expensive electron-phonon calculations for real materials and addressed them by the creation of artificial α 2 F's with modeling of simple physically motivated form, and the use of autoencoders, both equally good for learning Eliashberg theory.We also discussed future directions, including structure prediction utilizing surrogate machine learning models of energy landscapes to inform studies of metastable materials synthesized under high pressure, and extension of current methods to unconventional superconductors.

Figure 1 .
Figure 1.Performance comparison between the Allen-Dynes equation [2] and a model fit with symbolic regression alongside a proposed workflow for data augmentation.a) Allen-Dynes performance on low-TC, hydrides, and bimodal Einstein data.Plots are shaded by the log-density of points.b) Performance of symbolic regression machine learning model.c) Workflow for training an autoencoder neural network to learn an efficient compression of α 2 F. The mismatch between the original input and the reconstructed output, after compression and decompression, is minimized iteratively.d) The latter half of the trained autoencoder, known as the decoder, generates new α 2 F(ω) spectra by sampling the learned distribution.