Colloquium: Room temperature superconductivity: The roles of theory and materials design

For half a century after the discovery of superconductivity, materials exploration for better superconductors proceeded without knowledge of the underlying mechanism. The 1957 BCS theory cleared that up. The superconducting state occurs due to strong correlation in the electronic system: pairing of electrons over the Fermi surface. Over the following half century a higher critical temperature $T_c$ was achieved only serendipitously as new materials were synthesized. Meanwhile, the formal theory of phonon-coupled superconductivity at the material-dependent level became progressively more highly developed: by 2000, given a known compound, its value of $T_c$, the corresponding superconducting gap function, and several other properties of the superconducting state became available independent of further experimental input. In this century, density-functional-theory-based computational materials design has progressed to a predictive level; new materials can be predicted from free energy functionals on the basis of various numerical algorithms. Taken together these capabilities enable theoretical predictions for new superconductors, justified by applications to superconductors ranging from weak to strong coupling. Limitations of the current procedures are discussed; most of them can be handled with additional procedures. Here the process that has resulted in the three new highest temperature superconductors is recounted, with compressed structures predicted computationally and values of $T_c$ obtained numerically that have subsequently been confirmed experimentally: the designed superconductors ${\mathrm{SH}}_3$, ${\mathrm{LaH}}_{10}$, and ${\mathrm{YH}}_9$. These hydrides have $T_c$ in the 200–260 K range at megabar pressures; the experimental results and confirmations are discussed. While the small mass of hydrogen provides the anticipated strong coupling at high frequency, it is shown that it also enables identification of the atom-specific contributions to coupling, in a manner that was previously possible only for elemental superconductors. The following challenge is posed: that progress in understanding of higher $T_c$ is limited by the lack of understanding of screening of H displacements. Ongoing activities are mentioned and current challenges are suggested, together with regularities that are observed in compressed hydrides that may be useful to guide further exploration.

Figure 1

Plot of the major advances in the maximum $T_c$ vs year from the discovery of superconductivity in Hg in 1911 and using linear scales. Advances within classes are denoted by thick linear areas, with color delineating different classes. Certain temperature hurdles are denoted by dotted horizontal lines. High-pressure hydrides lie at the upper right (yellow), with temperatures converging toward room temperature. From [83].

Figure 2

Illustration of the predicted temperature dependence of the ratio of the specific heat in the superconducting state to that in the normal state value (equal to 1.0 on the abscissa). The behavior is exponentially small at low temperature due to the energy gap, which is in good agreement with experimental data. From [12].

Figure 3

McMillan’s plot for $5d$ transition metal alloys, of the experimental values (dots) of the band structure Fermi level densities of states, for which the mass enhancement $1+\lambda$ of the specific heat constant $\gamma$ has been removed, compared to the band theory calculation of tungsten by L. F. Matthiess. The agreement is noteworthy. From [70].

Figure 4

Bergmann and Rainer’s plots of $\delta T_c/\delta {\alpha }^{2}F(\omega )$ vs the ratio $\omega /T_c$ for a number of superconductors (as labeled). Its value peaks near $\omega =2\pi T_c$ and decreases slowly thereafter, indicating little importance of low energy phonons and great importance for higher energies. From [16].

Figure 5

(a) Allen and Dynes’s plots of the ratio $T_c/⟨\omega ⟩$ vs $\lambda$ for the McMillan equation, and for truncated and exact numerical results from the Eliashberg equation. (b) Similar plots using data for several known strong coupled superconductors. Also shown is the exact solution for an Einstein model with frequency $⟨\omega ⟩=\mathrm{\Omega }$, with the deviation from the McMillan equation beginning at $\lambda \sim 1$. Here $⟨\omega ⟩$ is the conventional first moment of ${\alpha }^{2}F(\omega )/\omega$. From [7].

Figure 6

Flow chart for the ab initio (entirely empirical-information-free) calculation for superconducting density functional theory, describing electrons and nuclei in thermal equilibrium in a superconducting state. See the original publication, from which the definitions of various items and the full formalism can be obtained. From [65].

Figure 7

Top panel: $T_c$ calculated within SCDFT plotted vs the experimental value. The dotted line along the diagonal indicates perfect agreement. The different piecewise linear lines indicate the use of three different dielectric functions for the Coulomb repulsion. Results for the TF-FE choice are almost indistinguishable from the experimental value. Bottom panel: analogous plot for the zero temperature static energy gap (calculations done at 0.01 K), which can be measured in several experiments. All show extremely encouraging agreement; the Thomas-Fermi–Sham-Kohn (TF-SK) values are somewhat better than the TF-FE values compared to experiment. No information on the Mo superconducting gap (which is small) is available. From [67].

Figure 8

Calculated vs experimental values of $T_c$ in $Fm\overline{3}m$ ${\mathrm{LaH}}_{10}$ and ${\mathrm{LaD}}_{10}$. The measured values are shown as dots from [26], and as dots with crosses from [97]. The solid (red) lines are from an anisotropic Eliashberg calculation, and the dashed blue lines are from SCDFT. Both include anharmonic phonons within a stochastic self-consistent harmonic approximation. Adapted from [34].

Figure 9

Top row: for the ${\mathrm{LaH}}_n$ system, the computed convex hull, with specific calculated formation enthalpies with respect to elemental La and ${\mathrm{H}}_2$. The three top panels display pressures of 100, 150, and 300 GPa for the stoichiometries that were obtained in the search. ${\mathrm{LaH}}_{10}$ is the compound of most interest, marked with a light green (gray) background. Convex hulls often result in several compounds near the convex hull and may be obtained from hundreds or thousands of enthalpy calculations. Adapted from [34]. Bottom row: schematic diagram of the progression of the high $T_c$ search program. First, the thermodynamic stability is obtained from compounds on the convex hull. Second, the dynamic stability is studied and phonon spectra are obtained. Third, the DFT-Eliashberg calculation, EP coupling and the phonon spectrum, is made to obtain ${\alpha }^{2}F(\omega )$, and $T_c$ is obtained from the Eliashberg equation or from one of the fit expressions $T_c(\lambda ,{\mu }^{*},{\omega }_{\log },{\omega }_2)$. DFTB, DFPT, and SCDFT indicate separate DFT-based capabilities. Adapted from [35].

Figure 10

Structures of (a) ${\mathrm{SH}}_3$, (b) ${\mathrm{YH}}_6$, and (c) ${\mathrm{LaH}}_{10}$ illustrating the high symmetry and the progression toward sodalite- or clathrate-type structures with increased H content. Small circles indicate H atoms, while larger circles denote metal atoms. (d) Schematic ${\omega }_2–T_c$ phase diagram of calculated results for five compressed hydrides in the three crystal structure classes shown in (a)–(c). Note that pressure $\mathcal{P}$ is increasing in the upper left direction, the blue region (lower right region) denotes the low-pressure region of instability of the structures that are considered. Solid lines connect values of a compound changing with pressure. Pressure lowers $T_c$; as pressure increases, ${\omega }_2$ increases but $\lambda$ and $T_c$ decrease. Lowering pressure increases $T_c$ (and $\lambda$) but leads to dynamic instability. The trends suggest an “island of instability” (blue region) for compressed hydrides. (e) Pickard’s scatterplot in the $P\text{−}{T}_c$ plane of experimental (hexagons) and theoretical (circles) positions for several superconductors. The contours indicate values of the ad hoc figure of merit $S=T_c(\mathrm{K})/[T_{c,{\mathrm{MgB}}_2}(\mathrm{K}{)}^2+P(\mathrm{GPa}{)}^2{]}^{1/2}$, which focuses attention on the desirability of low pressure $P$ for higher $T_c$; see Sec. 12 of [19].

Figure 11

Color-coded plots of several spectral functions of ${\mathrm{SH}}_3$ at the three pressures indicated in the top three panels, all above the optimum pressure of 160 K. The main point is to illustrate the separation of S and H modes (the solid vertical line at 80 meV), allowing analysis distinct from their coupling strengths and contribution to $T_c$. The blue (solid) vertical line marks this separation, lying in the gap around 80 meV between S (left, low frequency) and H (right, high frequency) vibrational modes. Top three panels: phonon density of states $F(\omega )$ for each of the three denoted pressures. The main effect of pressure is to push some H modes to higher frequency. Bottom three panels: ${\alpha }^{2}F(\omega )$, ${\alpha }^{2}F(\omega )/\omega$, and the ratio ${\alpha }^2(\omega )={\alpha }^{2}F/F$ that indicates the mean coupling matrix elements’ strength at frequency $\omega$, respectively. The arrows indicate peaks of interest. The low frequency region is subject to numerical noise and carries little weight, so it has been cut out. From [84].

Figure 12

Atomic H quantities in the indicated superconducting compressed hydrides, obtained by taking advantage of the spectral separation of metal acoustic modes and H optic modes. The corresponding calculated values of $T_c$ are shown in Fig. 10. (a) $(\eta ,\lambda )$ and (b) $(\kappa ,\lambda )$ phase planes, indicating small and large relative variations. (c)–(f) Pressure variations of the indicated quantities showing the magnitudes and rates of variation with pressure. Units: $\lambda$, unitless; $\eta =N(0){\mathcal{I}}^2$ and $\kappa =M_{\mathrm{H}}{\omega }_2^2$, $\mathrm{eV}/{\AA }^2$; ${\mathcal{I}}^2$, ${\mathrm{eV}}^2/{\AA }^2$. The background colors (different shadings in gray scale) indicate the different types of plots; see the axis labels. Adapted from [84].