{"id":219712,"date":"2025-09-12T02:19:23","date_gmt":"2025-09-12T02:19:23","guid":{"rendered":"https:\/\/www.europesays.com\/us\/219712\/"},"modified":"2025-09-12T02:19:23","modified_gmt":"2025-09-12T02:19:23","slug":"the-maximum-tc-of-conventional-superconductors-at-ambient-pressure","status":"publish","type":"post","link":"https:\/\/www.europesays.com\/us\/219712\/","title":{"rendered":"The maximum Tc of conventional superconductors at ambient pressure"},"content":{"rendered":"

Based on the observation that the coupling constant depends mainly on the phonon frequencies, McMillan in 196836<\/a> already derived an approximate expression of \\({T}_{{{\\rm{c}}}}^{\\max }\\). His result, however, only holds for a given class of materials and therefore does not provide an absolute value for this quantity. Further insight can be obtained by maximizing Tc given by Allen-Dynes\u2019 version of McMillan\u2019s formula37<\/a><\/p>\n

$${T}_{{{\\rm{c}}}}^{{{\\rm{McMillan}}}}=\\frac{{\\omega }_{\\log }}{1.20}\\exp \\left(-1.04\\frac{1+\\lambda }{\\lambda -{\\mu }^{*}(1+0.62\\lambda )}\\right),$$<\/p>\n

\n (1)\n <\/p>\n

where \u03bc* is the Coulomb pseudopotential, \u03bb is the electron-phonon coupling constant<\/p>\n

$$\\lambda=2 \\int_{0}^{+\\infty }\\frac{{\\alpha }^{2}F(\\omega )}{\\omega }d\\omega,$$<\/p>\n

\n (2)\n <\/p>\n

and \u03b12F(\u03c9) is the Eliashberg spectral function, calculated from the electron-phonon coupling matrix elements. With \\({\\omega }_{\\log }\\) we indicate the logarithmic average of the phonon frequencies:<\/p>\n

$${\\omega }_{\\log }=\\exp \\left[\\frac{2}{\\lambda }\\int_{0}^{+\\infty }\\ln (\\omega )\\frac{{\\alpha }^{2}F(\\omega )}{\\omega }d\\omega \\right].$$<\/p>\n

\n (3)\n <\/p>\n

By setting, for simplicity, the Coulomb pseudopotential \u03bc* to zero, one obtains that the maximum is attained at \u03bb\u00a0\u2248\u00a02. However, McMillan\u2019s formula is only valid for values of \u03bb \u2272 1.5, while no maximum value for Tc exists in the original Eliashberg theory29<\/a>. It is also frequently argued that the value of \u03bb is limited, as the lattice becomes eventually unstable for very large values of the coupling constant. However, there are some experimentally known superconductors with very high values of \u03bb, well above 2.0, both at ambient (e.g., Pb\u2013Bi compounds38<\/a>) and under pressure39<\/a>.<\/p>\n

Recent works estimated the maximum Tc of conventional superconductivity from fundamental limits40<\/a>,41<\/a>,42<\/a>. They all agree on a value of 300\u2013600 K at ambient pressure, suggesting that superconductors may exist at ambient temperature. In the following, we critically look at the basic assumptions used in these estimates. This is done by analyzing experimentally-known superconductors and our calculations of electron-phonon interaction and superconducting properties for more than 20 000 metals43<\/a>,44<\/a>. This is by far the largest dataset available to date with calculated superconducting properties, and it contains metals with a large variety of chemical elements and crystal structures. It also includes most of the materials that have been proposed in the literature as conventional high-Tc superconductors. These compounds had to be added manually, as in the overwhelming majority of the cases they were thermodynamically highly unstable and were therefore absent from our database. The \u03c3-electron systems that we have considered include a series of materials based on MgB2 structure45<\/a>,46<\/a>,47<\/a>,48<\/a>,49<\/a>,50<\/a>,51<\/a>,52<\/a>,53<\/a>,54<\/a>,55<\/a>,56<\/a>, on the diamond structure57<\/a>,58<\/a>,59<\/a>,60<\/a>,61<\/a>,62<\/a>, cage-like structures63<\/a>,64<\/a>,65<\/a>,66<\/a>,67<\/a>,68<\/a>,69<\/a>,70<\/a>,71<\/a>,72<\/a>,73<\/a>,74<\/a>, hydrides34<\/a>,75<\/a>,76<\/a>,77<\/a>, and many other compounds35<\/a>,78<\/a>,79<\/a>,80<\/a>,81<\/a>,82<\/a>. In several of these families, we also performed exhaustive machine-learning accelerated83<\/a> high-throughput calculations that went well beyond what had been reported in the literature. For example, we studied (with a X and Y running over the periodic table) all possible XY2 compounds in the structure of MgB2; XBC in the structure of LiBC55<\/a>; MgXB4 compounds in the structure of MgAlB456<\/a>; all possible B\u2013C phases in the diamond structure for unit cells with up to 12 atoms, all possible LixByCz structures with up to 12 atoms in the unit cell, all XB3N3 and XYB6C6 filled sodalite compounds64<\/a>, etc. Finally, this ab initio data was complemented by the investigation, using the machine-learning models developed in refs. 43<\/a>,44<\/a>, of the superconducting properties of more than 100 million compounds. While no study can be truly exhaustive given the vastness of chemical space, our datasets are comprehensive and extensive, providing us with a thorough overview of conventional superconductivity across materials space.<\/p>\n

To estimate the maximum Tc of conventional superconductors at ambient pressure, the first step is to determine what is the largest phonon frequency \\({\\omega }_{\\max }\\) that one can reasonably expect in a realistic compound. Reference42<\/a> derived the expression<\/p>\n

$${\\omega }_{\\max }=\\frac{{E}^{{{\\rm{electronic}}}}}{\\sqrt{m}},$$<\/p>\n

\n (4)\n <\/p>\n

where Eelectronic is a typical electronic energy, and m is an atomic mass. By inserting a typical Eelectronic\u00a0=\u00a01\u2009Ry, and the proton mass m\u00a0=\u00a0mproton into the formula, they obtained \\({\\omega }_{\\max }=3680\\) K. Compared to the vibrational frequency of H2 that is almost 6000\u2009K84<\/a>, this value does not appear to be an overestimation.<\/p>\n

The same conclusion can be reached by looking at the calculated values for our materials depicted in Fig.\u00a01<\/a>. The distribution of values of \\({\\omega }_{\\max }\\) exhibits two maxima, one at around 500\u2009K and another around 2000\u2009K. Although the overwhelming majority of the compounds have \\({\\omega }_{\\max }\\) below 3000\u2009K, we find a few that extend to 5400\u2009K, well beyond the estimation of ref. 42<\/a>. Not surprisingly, all of these are hydrides. An example of one such material, with a calculated \\({\\omega }_{\\max }=5396\\) K, is the hypothetical AgTl2H2. This is a compound that contains isolated H2 molecules inside a AgTl2 framework (see Supplementary Information, SI<\/a>). The H\u2013H distance is 0.78\u2009\u00c5, only slightly larger than the value of 0.76\u2009\u00c5 in solid hydrogen calculated in the same approximation. As expected by the extreme difference of masses, the phonon band structure splits into separated manifolds, with the lowest lying states with Tl and Ag character and the highest states coming from H. The H2 stretching mode has negligible dispersion, and can be found at around 5400\u2009K. The Ag\u2013Tl and lowest lying H-modes all couple very strongly to the electrons, leading to the large \u03bb\u00a0=\u00a01.1 (and a Tc\u00a0~\u00a011.5\u2009K). As expected by the very high frequency, the highest lying phonon mode has a negligible contribution to \u03bb, even if \u03b12F(\u03c9) exhibits a very high peak at that frequency.<\/p>\n

Fig. 1: Histogram of el-ph coupling properties.<\/b>\"figure<\/a><\/p>\n

Histogram of (top) the maxima (\\({\\omega }_{\\max }\\)) and logarithmic average (\\({\\omega }_{\\log }\\)) of phonon frequencies and (bottom) the electron-phonon coupling constants (\u03bb) calculated for around 20,000 metals (see ref. 43<\/a> for computational details).<\/p>\n

In Fig.\u00a01<\/a> we also plot the distribution of the values of \\({\\omega }_{\\log }\\). Contrary to \\({\\omega }_{\\max }\\), \\({\\omega }_{\\log }\\) has a single peak at very low frequency, and decays rapidly with frequency. At the end of the tail, at \\({\\omega }_{\\log }\\) values in excess of 1800\u2009K we find a few hydrides, such as the hypothetical perovskite NaNiH3, and some ordered crystals of boron-doped sp3 carbon (see\u00a0SI<\/a>). In the case of the hydride, only the high frequency H-modes have a significant coupling to the electrons, leading to a large logarithmic average but a very small value of \u03bb and consequently of Tc. The boron-doped case takes advantage of the high-energy of the carbon modes (due to the very strong C\u2013C sp3 bond) and from the fact that phonon modes couple strongly to the electrons in a large energy range. As expected, also in this case \u03bb has moderate values, of the order of 0.5\u20130.6, leading to Tc in the range of 10\u201330\u2009K.<\/p>\n

As the determinant factor for the calculation of Tc is \\({\\omega }_{\\log }\\) and not \\({\\omega }_{\\max }\\), from this discussion it seems much more reasonable to use values of the order of 1800\u2009K, and not \\({\\omega }_{\\max }=\\, 3680\\) K, as in ref. 42<\/a>.<\/p>\n

The next step in the estimation of the maximum value of Tc is the optimization of the shape of \u03b12F(\u03c9), assuming a maximum phonon frequency of \\({\\omega }_{\\max }\\). A free optimization of this function would obviously lead to Tc\u00a0=\u00a0\u221e, as \u03b12F(\u03c9) is not constrained by any sum-rule. Therefore, Trachenko et al. fixed \u03bb\u00a0=\u00a0242<\/a>, the value that maximizes Tcaccording to McMillan\u2019s formula. From the considerations above, and from the lower panel of Fig.\u00a01<\/a>, \u03bb\u00a0=\u00a02 seems to be a reasonable value, perfectly reachable in a variety of compounds. The hypothetical compounds we have in our dataset with highest values of \u03bb\u00a0\u2248\u00a03.3 are ClB2C8 and Al2OsH7 (see\u00a0SI<\/a>). The former is a C clathrate p-doped with B and co-doped with endohedral Cl, with a \\({\\omega }_{\\log }=425\\) K and a calculated Tc\u00a0=\u00a055 K. The latter, that exhibits a Al2Os framework that includes a large quantity of hydrogen, has an \\({\\omega }_{\\log }\\) of almost 300\u2009K, leading to Tc\u00a0=\u00a038\u2009K.<\/p>\n

The optimal shape of \u03b12F(\u03c9) obtained by Ref. 42<\/a> by optimizing the Tc calculated from the Eliashberg equations is a narrow peak at \\({\\omega }_{\\max }\\). Actually, the same conclusion follows directly from McMillan\u2019s formula for Tc. For a fixed value of \u03bb, the value of Tc changes linearly with \\({\\omega }_{\\log }\\). In turn, the maximum value of \\({\\omega }_{\\log }\\) is obtained when \\({\\alpha }^{2}F(\\omega ) \\sim \\delta (\\omega -{\\omega }_{\\max })\\), leading to \\({\\omega }_{\\log }=\\, {\\omega }_{\\max }\\). If, for the sake of the argument, we insert \u03bb\u00a0=\u00a02, \\({\\omega }_{\\log }=1800\\) K into the McMillan function, and assume a value of \u03bc*\u2009=\u20090.1, we obtain Tc\u00a0=\u00a0260 K. This is smaller than the value of ref. 42<\/a>, but still much larger than the current record of MgB2.<\/p>\n

It is most likely, however, that this limit is unattainable for any physical system at ambient pressure. In fact, the parameters \\({\\omega }_{\\log }\\) and \u03bb are not entirely independent, as they are two different moments of the same \u03b12F(\u03c9) function. This was already recognized in the seminal work of McMillan in 196836<\/a>, where it was shown that \u03bb is inversely proportional to the square of the average phonon frequency. Obviously \\({\\omega }_{\\log }\\) is favored by high frequencies, while \u03bb by low frequencies, so the shape of the optimal \u03b12F(\u03c9) that maximizes Tc (and that only contains a single, very high-frequency, flat-band phonon mode that couples very strongly to the electrons) seems to be unattainable physically.<\/p>\n

This reasoning is also supported by the data depicted in Fig.\u00a02<\/a>, where we plot the relationship between \\({\\omega }_{\\log }\\) and \u03bb for all compounds in our dataset. The size of the circles is proportional to Tc and in the background we plot the contour lines of constant Tc as calculated from McMillan\u2019s formula with \u03bc*\u00a0=\u00a00.1. We see that, as expected, compounds with very large values of \\({\\omega }_{\\log }\\) often have small \u03bb and vice-versa. Furthermore, no material comes close to the optimal case, and compounds with the highest values of Tc are the ones that achieve a good compromise between reasonable high \\({\\omega }_{\\log }\\) and \u03bb.<\/p>\n

Fig. 2: Scatter plot of \\({\\omega }_{\\log }\\) versus \u03bb for all calculated systems.<\/b>\"figure<\/a><\/p>\n

The size of the circles is proportional to Tc calculated from Allen-Dynes formula with correction factors37<\/a> and \u03bc*\u2009=\u20090.1. The contour lines are included as a guide to the eye and are obtained from McMillan\u2019s formula of eq. (1<\/a>) with \u03bc*\u00a0=\u00a00.1.<\/p>\n

As an example, we will look at the compounds with the highest Tc in our dataset, specifically Li2AgH6 and Li2AuH6 (see Fig.\u00a03<\/a> and Fig.\u00a0S1<\/a> in\u00a0SI<\/a>). These compound crystallize in the same cubic structure as Mg2IrH6, Mg2PdH6, Mg2PtH6, etc. that were recently proposed85<\/a>,86<\/a>,87<\/a> as high-Tc superconductors. Both materials are thermodynamically unstable at ambient pressure (see Fig.\u00a0S2<\/a> in\u00a0SI<\/a>) at respectively 0.319\u2009eV\/atom and 0.172\u2009eV\/atom above the convex hull for the Ag and the Au compounds, and are not significantly stabilized by pressure (at least up to 50 GPa, see\u00a0SI<\/a>).<\/p>\n

Fig. 3: Phonon and electron phonon coupling spectra of Li2AgH6.<\/b>\"figure<\/a><\/p>\n

From left to right, the phonon band structure, density of phonon states, and Eliashberg spectral function \u03b12F(\u03c9), calculated within the harmonic approximation.<\/p>\n

The electronic band structure of Li2AgH6 resembles the one of Mg2IrH6, with a single band crossing the Fermi surface. This band, with very strong H-character, is however more dispersive in the present case. Also the phonon band structure is similar, with the acoustic modes mostly composed by vibrations of the heavier atom, followed by modes stemming from the alkali or alkaline earth metal. Finally, there are three separate manifolds of phonon bands. In Li2AgH6 the lowest manifold contains both H and Li character due to the small difference between the atomic masses of these elements. Essentially all modes contribute to the very high value of \u03bb\u00a0\u2248\u00a04.<\/p>\n

We would like to note that one of the acoustic phonon branches exhibits (small) imaginary values close to the \u0393. This is a shortcoming of the harmonic approximation, and the structure is perfectly dynamically stable when anharmonic and quantum nuclear effects are taken into account (see\u00a0SI<\/a>)88<\/a>,89<\/a>. In any case, 95% of the electron-phonon coupling comes from mid-low frequency modes which are almost identical between harmonic and anharmonic calculations.<\/p>\n

In order to obtain a more accurate Tc estimation within modern superconductivity methods we have recomputed the value of Tc using two state-of-the art approaches which, unlike more conventional methods, include the electron phonon coupling and the Coulomb interaction calculated from first-principles. These are the full Eliashberg approach of Ref. 90<\/a> and density functional theory for superconductors91<\/a>,92<\/a>,93<\/a>,94<\/a>,95<\/a> (SCDFT). For the sake of these high-accuracy calculations the electron-phonon coupling was recomputed using a Monte-Carlo k-mesh accumulated on the Fermi surface, on which the electron-phonon matrix elements are linearly interpolated96<\/a>. This ensures a perfect convergence of the nesting properties entering the definition of the Eliashberg spectral function.<\/p>\n

The resulting critical temperature for Li2AgH6 is 108.8\u2009K in Eliashberg theory and 83.0\u2009K in SCDFT. Although both approaches are derived from the Migdal approximation for the electron-phonon self energy97<\/a> and assume a static Coulomb interaction, the slightly different Tc prediction is dictated by the approximation chain that leads to each computational scheme. On one hand, in SCDFT there is the approximation to the anomalous exchange-correlation functional92<\/a>, which adds additional approximations to the form of the self-energy. On the other hand, in Eliashberg theory the energy dependence of the Coulomb interaction is neglected at the scale of the phononic energies. However, considering that the electron-phonon coupling parameter \u03bb is very high, the functional approximation of SCDFT might be slightly beyond its validity range92<\/a>. Therefore we expect that, in this case, the Eliashberg estimation should be more accurate.<\/p>\n

It is interesting that the Eliashberg estimation of Tc is in very good agreement with the one obtained by means of the McMillan-Allen-Dynes approach using a standard \u03bc*\u00a0=\u00a00.1. This indicates that Coulomb interactions act as expected for a conventional sp-system98<\/a>. This differs from what is observed in Mg2IrH6 which, as discussed in ref. 85<\/a>, has its Tc overestimated by the \u03bc* model99<\/a>. The reason is the presence of a peak in the density of states at the Fermi level, that is close to a large band gap, leading to a poor Coulomb renormalization98<\/a>,100<\/a>. The Li2AgH6 system also features a peak in the density-of-states and a band gap, however the peak is broader, while the band gap is very small, leading to an overall smoother density profile and efficient Coulomb renormalization.<\/p>\n

A similar analysis can be extended to Li2AuH6. This system has electronic, phononic and superconducting properties nearly identical to its Ag twin. The predicted superconducting Tc including Coulomb interactions is 91.0\u2009K and 116.1\u2009K in SCDFT and Eliashberg, respectively.<\/p>\n

In view of the discussion before, these materials seem to be ideal cases for conventional superconductivity, and their Tc is likely in the maximum range of what can be achievable at ambient pressure. We note that these are isotropic superconductors, where the effect of anisotropy accounts for less than 1% of the value of Tc. This is the opposite of MgB2, where the electron-phonon coupling mostly acts on the \u03c3 bands, and an isotropic calculations leads to an underestimation of Tc by almost a factor of two. The isotropic superconducting state of Li2AgH6 or Li2AuH6, if experimentally realized, would not only have a critical temperature above liquid nitrogen, but it would also be more suitable for high-field applications. In fact these applications require impurities and defects to work as pinning centers for magnetic flux lines and reduce the Ginsburg-Landau coherence-length. Isotropic superconductors are more likely to be stable upon the introduction of crystalline defects and still be affected by their scattering potential.<\/p>\n

At this point it it important to recognize the approximations used in our workflow, and discuss their potential impact on the maximum value of Tc.<\/p>\n

Finite size of the dataset<\/p>\n

By using our machine learning model we have by now screened more than 100 million compounds, including almost 3 million hydrides. For all compounds predicted by the machine learning model to have a Tc above 10 K, we have performed full electron-phonon calculations. We also performed successive cycles of training the model and predicting Tc, creating an iterative refinement process that has exhaustively explored the chemical space. By now, the error of our machine learning model has converged to below 1 K, indicating that we have captured the essential physics of electron-phonon coupling in these materials. Furthermore, all our recent batches consistently fail to find any new superconductors with very large Tc (>50\u2009K), even as our material database grows at an impressive rate of 1\u20132 million compounds per week. This plateau in discoveries, despite exponentially increasing data, provides strong statistical evidence that we have identified the fundamental upper limits of superconducting transition temperatures.<\/p>\n

Anisotropy and multi-band effects<\/p>\n

We have been systematically performing anisotropic calculations, either by solving the anisotropic Eliashberg equations or through density-functional theory for superconductors. For all compounds that appear at the top of our Tc list, anisotropic effects turn out to be very minor, and do not affect Tc significantly. This points to the conclusion that our best compounds are already \u201cfully optimized\u201d, and Tc has indeed reached its maximum value. Of course, for compounds with a lower transition temperature there is a larger margin for optimization. This can be seen from, e.g., ref. 101<\/a>, where anisotropic calculations were performed for 242 materials, finding a significant increase in the Tc for a few compounds. The resulting Tc values, however, are still well below our suggested maximum Tc of around 100 K, in agreement with our arguments.<\/p>\n

Doping<\/p>\n

Although this is a crucial strategy to improve superconducting properties we evaluate that it will not significantly affect our statistical analysis of the maximum Tc. Let us divide this discussion in two parts: a) standard doping, where a small percentage of dopant atoms is added, removed or substituted, in an attempt to move the Fermi level to a region of higher density of states or of softening some phonons to a sweet spot in frequency. In a first approximation this can be achieved by a rigid shift of the Fermi energy in the calculations. We have tried this approach for our best candidates with partial success (see, e.g., Mg2PdH6 and Mg2PtH685<\/a>). In fact, materials with high critical temperature are usually already in a near-optimal configuration, and in this case their Tc can not be significantly improved by doping. This is the case of MgB2102<\/a> or Mg2IrH685<\/a>, for example. Of course, other compounds with lower values of Tc have a larger potential for optimization, but their optimized Tc remains moderately low. b) degenerate doping, where a significant amount of atoms is added, removed or substituted. Indeed this often leads to substantial increase of Tc, as it can be seen, for example, in Ref. 69<\/a>. However, this kind of systems are to a large extent already represented in our dataset, and should therefore be easily predicted by our machine-learning model. While our analysis focuses on ordered crystalline compounds, we note that the effect of alloying or disorder in doped systems remains a challenging computational problem that has been addressed in only a limited number of studies103<\/a>,104<\/a>.<\/p>\n

Anharmonic effects<\/p>\n

Phonon anharmonic effects can be crucial for many systems. However, their primary impact lies in the stabilization of otherwise unstable or marginally stable phases. This stabilization mechanism is exemplified in H3S, where anharmonic effects enable the cubic phase to become stable at lower pressures than predicted by the harmonic approximation105<\/a>. Furthermore, the stabilization of soft-modes usually leads to a decrease of \u03bb and consequent decrease of Tc as, for example, in H3S under high pressure105<\/a> or in PdH at ambient pressure106<\/a>.<\/p>\n

Reduced dimensionality<\/p>\n

Recently some of us applied our approach to 2D systems107<\/a>, motivated by the hypothesis that reduced dimensionality could enhance transition temperatures. However, the values of Tc for 2D systems turned out to be systematically lower than for 3D compounds. In retrospect, this outcome can be understood through fundamental physical principles. First, 2D systems exhibit systematically lower electronic densities of states at the Fermi level compared to 3D materials. Second, the frequencies of out-of-plane phonon modes are characteristically reduced in layered structures. While this frequency reduction does lead to an increase in the electron-phonon coupling parameter \u03bb, it simultaneously causes a substantial decrease in the logarithmically averaged phonon frequency \\({\\omega }_{\\log }\\). According to the McMillan formula, the net effect of these competing factors results in an overall reduction of Tc, despite the enhanced coupling strength.<\/p>\n

Treatment of the Coulomb interaction<\/p>\n

Most often the Coulomb interaction is handled at the level of \u03bc* approximation, and this is also the case for the data in Figs.\u00a02<\/a> and 4<\/a>. The standard value of \u03bc*\u00a0=\u00a00.1 (that we also adopt) is likely a lower limit for real systems, and it is often a gross underestimation for compounds with very high values of Tc. For example, for Mg2IrH685<\/a> we computed the Coulomb interaction from first principles, and found that the resulting repulsion is extremely large at the Fermi level (yielding a \u03bc\u2009=\u20090.58), consistent with the large electronic density of states. Furthermore, as the density of states decreases away from the Fermi level, the Coulomb renormalization mechanism becomes very inefficient. A direct calculation of \u03bc* is quite difficult, but a rough estimation indicates that the value should be \u03bc* \u226b 0.25, much larger than the 0.1 value commonly used, hence leading to a considerable decrease of Tc.<\/p>\n

Fig. 4: Scatter plot of the distance to the convex hull of stability versus the superconducting transition temperature for all compounds in our dataset.<\/b>\"figure<\/a><\/p>\n

In orange we also indicate the Pareto front corresponding to this data. Compounds on the Pareto front are labeled. Values of Tc care alculated from Allen-Dynes formula with correction factors37<\/a> and \u03bc*\u00a0=\u00a00.1.<\/p>\n

Finally, we would like to discuss the issue of synthesizeability. Most of the compounds discussed above, and others that have been proposed in the literature, with high-Tc are hypothetical, and up to now none has been stabilized at ambient pressure. To better understand this problem, we plot in Fig.\u00a04<\/a> the distance to the convex hull of thermodynamic stability and the calculated transition temperature of all compounds in our dataset. We also indicate the Pareto front that corresponds to this data. The values of Tc are obtained with an isotropic theory, so the transition temperature of MgB2 is considerably underestimated. We also note that although being close to the hull is not a synonym of synthesizeability, the higher the distance to the hull for a given compound, the smaller the probability that this compound can be stabilized experimentally.<\/p>\n

Very close to the hull, we find that the compound with highest predicted Tc is LiMoN2 with around 40\u2009K44<\/a>. Unfortunately, this compound exhibits intrinsic defects that lower the high density of states at the Fermi level and destroy superconductivity44<\/a>,108<\/a>. If it is possible to resolve this problem experimentally and synthesize the pristine compound is at this point unknown109<\/a>. At higher values of Tc we find several hydrides and boron-carbides. Unfortunately, all these compounds are unstable thermodynamically with a distance to the hull that increases rapidly along the Pareto front. This is easy to understand as hydrides prefer charge-compensated, semiconducting (or insulating) phases and are therefore destabilized in the metallic phase, and boron induces stresses in the very strong diamond framework increasing its energy. A possible path through the high-pressure synthesis followed by quenching of these compounds to ambient pressure is commonly suggested in the literature110<\/a>,111<\/a>,112<\/a>. This approach was recently used to quench a superconducting phase of the topological alloy compound Bi0.5Sb1.5Te3113<\/a> from 4\u2009GPa to ambient pressure with Tc reaching 10\u2009K. Also FeSe has been subject to pressure quenching from 4.15\u2009GPa reaching a Tc of 37\u2009K at ambient pressure114<\/a>, as well as Sb, quenched from 10.9 GPa and retaining a Tc of 3\u2009K at ambient pressure. Another example is the clathrate SrB3C3, synthesized under high pressure and measured to be a superconductor with a maximum Tc of 22\u2009K at 23\u2009GPa68<\/a>. In the latter case, it was argued that the sample survived to ambient pressure, but it was not possible to measure superconductivity. Despite these successes, the possibility of quenching superconducting hydrides (with extremely mobile hydrogen atoms) from the very high pressures required for their synthesis to ambient conditions is still unproved to this date, and remains highly speculative.<\/p>\n

In summary, by analyzing data from more than 20,000 electron-phonon calculations we critically discussed the possible maximum Tc of conventional superconductors at ambient pressure. It seems clear that it is possible to design hypothetical compound with values of Tc reaching 100\u2013120\u2009K, much larger than the current experimental record, but still very far from room temperature. Unfortunately, all compounds with high Tc appear to be thermodynamically unstable, raising questions about their experimental synthesis and characterization. It is true that physical laws do not restrict Tc to go beyond 100\u2013120\u2009K, but in practice our data show that the experimental realization of a compound with such high Tc is extremely unlikely.<\/p>\n

However, we do not exclude that room-temperature superconductivity could be achieved through unconventional pairing mechanisms, high pressure conditions, or entirely new physical phenomena that transcend current theoretical frameworks. Our hope is that this analysis helps guide future research toward the most promising avenues while establishing realistic expectations about the fundamental limits of conventional electron-phonon mediated superconductivity.<\/p>\n","protected":false},"excerpt":{"rendered":"Based on the observation that the coupling constant depends mainly on the phonon frequencies, McMillan in 196836 already…\n","protected":false},"author":3,"featured_media":219713,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[25],"tags":[10046,10047,492,159,27577,67,132,68],"class_list":{"0":"post-219712","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-physics","8":"tag-humanities-and-social-sciences","9":"tag-multidisciplinary","10":"tag-physics","11":"tag-science","12":"tag-superconducting-properties-and-materials","13":"tag-united-states","14":"tag-unitedstates","15":"tag-us"},"share_on_mastodon":{"url":"https:\/\/pubeurope.com\/@us\/115188935675359581","error":""},"_links":{"self":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/219712","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/comments?post=219712"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/posts\/219712\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media\/219713"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/media?parent=219712"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/categories?post=219712"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/us\/wp-json\/wp\/v2\/tags?post=219712"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}