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Predicting superconductors

56 sources, listed at the end · last revised 7 October 2026

Predicting superconductors means calculating, before a material is made, whether it will superconduct and at what transition temperature (T_c). The calculation is possible for conventional superconductors, in which electrons pair through vibrations of the crystal lattice, called phonons. A theory formulated in 1960 gives their T_c from quantities that can be computed from the crystal structure.^[1] The hydrides (hydrogen-rich compounds) H₃S and LaH₁₀, which superconduct at 203 K (−70 °C) and about 250 K (−23 °C) under 155 and about 170 gigapascals (GPa), more than 1.5 million atmospheres, were calculated before the measurements were published.^[2][3][4][5]

The same calculation has since been run for more than 20,000 metals at ambient pressure,^[6] and machine-learning models trained on such results or on measured transition temperatures have ranked candidates, in one case more than 1.3 million structures.^[8] The new superconductors found by making and measuring compounds that such models proposed all have T_c below 10 K.^{[7][8][9][10][12][13]}

The compounds with the highest T_c calculated for ambient pressure are unstable against decomposition, and some other predicted compounds did not form in synthesis attempts.^[6][14][15] For some classes of unconventional superconductors, among them the cuprates that hold the ambient-pressure record of 133 to 138 K, no quantitative theory of T_c exists.^[11][16][17] Whether phonon pairing can reach room temperature at ambient pressure is disputed.

Why conventional superconductors can be calculated

In a conventional superconductor the attraction that binds electrons into pairs comes from phonons, the vibrations of the crystal lattice (see Superconductivity). Migdal-Eliashberg theory, formulated in 1960, turns that picture into equations that can be solved for T_c.^[1]

The equations take as input a spectrum that records how strongly the phonons of each frequency couple to the electrons. Two numbers summarize it. The electron-phonon coupling strength, written λ, is a dimensionless measure of the attraction the phonons provide.^[1] In the preprint of a calculated survey by Cerqueira and coauthors, its mean over 6,912 stable or nearly stable metals was 0.37, and a distribution fitted to the values puts about 1% of such compounds above 1.^[18] For H₃S at 200 GPa the calculated value is 2.64 if the atoms are assumed to vibrate like ideal springs (the harmonic approximation) and 1.84 without that assumption, giving T_c values of 250 K and 194 K against 190 K measured at that pressure.^[19] The characteristic phonon frequency, a logarithmic average written ω_log, is quoted as a temperature: its mean in the same preprint was 229 K, and Gao and coauthors, who surveyed more than 20,000 metals in 2025, report that it rarely exceeds 1,800 K.^[18][6] T_c rises with both quantities.^[1] A third number, the Coulomb pseudopotential μ*, stands for the repulsion between electrons, which works against pairing, and is usually set by convention between 0.1 and 0.2.^[1][6][20]

Two formulas estimate T_c from these numbers without solving the full equations. McMillan's formula, published in 1968, was fitted to numerical solutions for λ up to about 1. The Allen-Dynes formula of 1975 uses ω_log and adds corrections for stronger coupling and for the shape of the spectrum.^[21][22][1]

For the superconducting elements, calculated values of T_c usually lie within 20% of the measured ones.^[23] A benchmark that computed the repulsion between electrons, and so contained no adjustable number, reproduced the measured values except for zinc, cadmium and vanadium.^[24] The simplified treatment used in large surveys, which gives all the conduction electrons the same pairing strength, can be further off: for magnesium diboride, MgB₂, measured at 39 K (−234 °C), it gives 18.9 K.^[18][6][25]

The coupling of each kind of atom can be written as an electronic factor, the Hopfield parameter, divided by the atom's mass and the square of its vibration frequency.^[26] In the limit of very strong coupling the frequency cancels, and the theory gives a T_c proportional to the square root of the Hopfield parameter divided by the mass.^[26][27] A 2025 paper by M. V. Sadovskii presents that value as an upper limit on T_c, and a 2019 analysis found that the high-pressure hydrides are not in that limit.^[27][26] The lab behind this site publishes an estimator of that ratio summed over the atoms of a structure (model).

From chemical formula to crystal structure

The inputs come from density functional theory (DFT), a quantum-mechanical method that computes the energy of a given arrangement of atoms. Phonons and their coupling to the electrons follow from how that energy changes when atoms are displaced.^[23] Crystal structure prediction programs search for the arrangement that a compound never made before should adopt: they generate trial arrangements at random or with evolutionary and particle-swarm algorithms, move each with DFT to the nearest energy minimum, and keep the most stable at the chosen pressure.^[23]

High-pressure hydrides predicted in this way and later confirmed by experiment include H₃S, LaH₁₀ and YH₉.^[1] A calculation by Duan and coauthors, published in November 2014, found that hydrogen sulfide with added hydrogen forms a cubic compound, H₃S, above 180 GPa, with T_c of 191 to 204 K at 200 GPa.^[2] An experiment published in 2015, prompted by an earlier prediction of about 80 K for H₂S, measured 203 K at 155 GPa and attributed the result to H₃S formed as the H₂S decomposed.^[3][23] For lanthanum decahydride, LaH₁₀, a 2017 calculation gave 274 to 286 K at 210 GPa, and a 2019 experiment found about 250 K at about 170 GPa.^[4][5] CaH₆, predicted in 2012 at 220 to 235 K and 150 GPa, was made a decade later and measured at 215 K and 172 GPa.^[28][29]

The 2017 calculation also gave 305 to 326 K at 250 GPa for YH₁₀, which did not form in experiments at pressures up to 410 GPa, and the yttrium hydrides that did form superconduct about 30 K below their predicted values.^[4][15] A 2020 review counted predictions of superconductivity for the hydrides of 61 elements at pressures up to 300 GPa; by the end of 2019 a transition had been measured under pressure for seven.^[23] A 2024 calculation predicted LaSc₂H₂₄ with T_c up to 331 K at 250 GPa.^[30] A 2025 preprint reports its synthesis and resistive onsets up to 298 K at 260 GPa. The result has not been reproduced, and a second group did not obtain the compound in seven attempts (see Claims of room-temperature superconductivity).^[31][32]

High-throughput screening and open databases

High-throughput screening applies the same calculation to every suitable entry of a database of crystal structures. The table gives two open databases of this kind and SuperCon, an open record of measured transition temperatures.

The Materials Project was started in 2011 at Lawrence Berkeley National Laboratory.^[33] Alexandria also distributes phonon and electron-phonon calculations,^[36] and the Gao survey deposited its data there.^[6] SuperCon is published by Japan's National Institute for Materials Science.^[37]

Few metals couple strongly, so a screen is built as a funnel: a fast model ranks a large set of candidates and the full calculation is run only on the best-ranked. Cerqueira and coauthors calculated the electron-phonon coupling of about 7,000 compounds, trained a model on the results to rank about 200,000 metals, and confirmed 541 with a calculated T_c above 10 K.^[38] Their preprint put the chance that a stable, non-magnetic metal exceeds 20 K at about 0.4%.^[18] A survey limited to compounds that have already been made examined 4,533 non-magnetic metals (magnetism works against this pairing) and predicted 24 unknown superconductors above 10 K.^[39]

Machine-learning models

Models trained on measurements take a chemical formula as input and learn from SuperCon. One, built on about 16,400 SuperCon compounds, about 4,000 of them with no reported T_c, sorted compounds held out of training into those above and below 10 K with about 92% accuracy.^[40] Models trained on calculations take a crystal structure and return λ, ω_log or the whole coupling spectrum. One is trained on about 7,000 calculated spectra and was used to rank more than 1.3 million structures.^[8]

Peer-reviewed papers report superconductivity in samples of at least six compounds proposed with the help of such models. Mo₂₀Re₆Si₄ superconducts at 5.4 K.^[7] Be₂HfNb₂ and Be₂Hf₂Nb showed resistive onsets at 4.24 and 3.18 K, against calculated values of 5.1 and 9.3 K, in samples that also held unidentified impurity phases, and the Be₂HfNb₂ sample showed signs of a second superconducting phase.^[8] YRu₃B₂ and LuRu₃B₂ superconduct at 0.81 and 0.95 K.^[9] YRu₃B₂ had been made in 1980 and tested then only down to 1.2 K, and another group measured its transition at 0.7 K.^[41] A synthetic sample of the mineral monchetundraite, Pd₂NiTe₂, shows magnetic shielding that sets in at 1.06 K and reaches about 5% of its volume, a result its authors say needs confirmation in purer samples.^[10]

Of 18 three-element compounds proposed by a generative model (one that outputs new crystal structures) and attempted in the laboratory, nine samples superconduct, between 4.8 and 9.7 K. The predicted crystal structures in general did not form, and the samples were mostly disordered alloys or mixtures of known phases.^[12] Another search found a superconducting signal near 9 K in samples of zirconium, indium and nickel without isolating the phase responsible.^[13] All of these transitions are below 10 K (−263 °C).

The first limit of these models is extrapolation. A model trained on one family of superconductors had no predictive power for the others; cuprates made up about 35% of its data.^[40] A generative model asked for structures with T_c of 110 K, above anything in its training data, did not produce them,^[12] and the authors of the spectrum model state that it is unlikely to predict high-temperature superconductors resembling the hydrides.^[8]

The second limit is duplication. About 10,000 SuperCon entries repeat a formula already present, and many others are one compound with slightly different amounts of an added element. A test set drawn at random therefore contains close relatives of the training compounds, and accuracy measured on it overstates how a model performs in discovery.^[35][40][42]

The third is the shortage of negative examples. Failed searches are rarely published, and an entry without a T_c may only mean that the sample was not cooled far enough.^[43][40] One group trained on synthetic negatives, made by assuming that the entries of a crystal-structure database that are absent from SuperCon do not superconduct.^[43]

Stability and synthesis

Thermodynamic stability is judged with the convex hull, the set of phases whose energy is lower than that of any combination of other phases with the same overall composition.^[23][44] A compound above the hull can lower its energy by decomposing into other phases, and persists as a metastable phase only where an energy barrier slows the decomposition. Of 29,902 known inorganic crystals about half are metastable. Among those the median height above the hull is 15 millielectronvolts (meV) per atom, and nine in ten lie within 67 meV per atom. The authors of that count argue that each observed metastable phase was the most stable phase under some condition met during its formation, so a hypothetical phase with no such condition may be unreachable.^[44] A structure must also be dynamically stable: any small displacement of the atoms raises the energy, so they return to their positions.^[23]

In the Gao survey the compounds with the highest calculated T_c, Li₂AgH₆ and Li₂AuH₆ at 83 to 116 K depending on the method, lie 319 and 172 meV per atom above the hull. Close to the hull, the compound with the highest calculated T_c is LiMoN₂, near 40 K. It was first made in 1992, and the survey's authors state that intrinsic defects in it destroy the superconductivity.^[6][18]

For Mg₂IrH₆, published calculations put T_c at ambient pressure between 65 and 170 K.^[14][45] Synthesis attempts up to about 28 GPa produced Mg₂IrH₅, an insulator with the same structure and one hydrogen atom missing.^[14] A 2026 preprint reports Mg₂IrH₇ above about 40 GPa, an insulator that reverts to Mg₂IrH₅ when the pressure is lowered to about 20 GPa.^[46] Neither study obtained Mg₂IrH₆.^[14][46]

Unconventional superconductors

The calculations above assume phonon pairing. For the iron-based compound LaFeAsO they allow a T_c of at most 0.8 K, against 26 K measured in the fluorine-doped compound.^[47][48] A 2016 review noted that no quantitative theory exists for some classes of known superconductors, and that the heavy-fermion compounds (metals containing rare-earth or actinide elements), the cuprates and the iron-based superconductors were each found by experiment without having been anticipated.^[11] The cuprates hold the highest T_c of a stable phase at ambient pressure, 133 to 138 K (−140 to −135 °C).^[16][17] A 2024 numerical study of the Hubbard model, a simplified description of the electrons in a cuprate layer, found a superconducting ground state and did not predict a transition temperature.^[49]

The first nickel oxide superconductor was found by analogy with the cuprates: nickel oxides were examined as possible analogs in 1999, and a thin film of one with the crystal structure of a cuprate was found to superconduct at 9 to 15 K in 2019.^[50][51]

Proposed limits on phonon-mediated superconductivity

MgB₂, found to superconduct in 2001 without having been predicted, was still the record phonon-mediated superconductor at ambient pressure in the Gao survey of 2025.^[25][11][6]

McMillan's formula levels off when it is extended to strong coupling, and many readers took it to show that theory limits T_c to around 40 K. Allen and Dynes showed in 1975 that the full equations set no maximum.^[1][22] In 2018 Esterlis, Kivelson and Scalapino proposed an approximate bound: T_c does not exceed about one tenth of the characteristic phonon frequency expressed as a temperature.^[52] A 2022 paper with Kivelson among its authors built models in which that ratio is unbounded and concluded that the bound is not fundamental, though useful in many practical cases.^[53] Semenok, Altshuler and Yuzbashyan argue that the equilibrium between electrons and lattice becomes unstable when λ exceeds about 4, and that phonon-mediated superconductivity at room temperature is feasible only in hydrogen compounds.^[54] Sadovskii argues that λ can become very large within a stable metal.^[27] Trachenko and coauthors derive from fundamental constants an upper limit on T_c of the order of 100 to 1,000 K, which does not exclude room temperature.^[55]

Gao and coauthors found that compounds with a high ω_log tend to have a small λ. They state that physical laws do not strictly cap T_c but call conventional superconductivity at room temperature and ambient pressure "extremely unlikely", and they add that no study of this kind can be exhaustive.^[6] A later calculation that shares two authors with the survey gives 133 K, with μ* fixed at 0.1, for cubic PdH₄ at ambient pressure. The paper reports that the structure is dynamically stable; its preprint gives no energy above the convex hull, and the compound has not been made.^[56]

See also

• Superconductivity
• High-pressure hydrides
• Superconducting materials
• Room-temperature superconductor
• Verifying a superconductor

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