Introduction

Since 1911, Onnes’s discovery of the superconductivity phenomenon of zero resistance in Hg, the continues efforts have been made to create and find a room temperature superconductor possessing an intriguing scientific and technological potential. Ashcroft predicted that the room-temperature Tc can be achieved for hydrogen solid metal with an extremely high Debye temperature given as inversely proportional to root hydrogen mass \({\upomega }_{Debye} \propto 1/\sqrt {M_{Hydrogen-mass} }\)1. In 1935, Wigner and Huntington claimed that at a pressure of 25 gigapascals (GPa), solid molecular hydrogen would turn into a metal2. Silvera and Dias managed to turn hydrogen to metallic at a pressure of 495 GPa, well beyond the 360 GPa of Earth’s core3. In 1970, Satterthwaite & Toepke first observed superconductivity of Tc ≈ 8.05 ~ 8.35 K in the hydrides and deuterides of thorium with H-or D-to-metal atom ratios of 3.60–3.654. They asserted that these materials are apparently type-II superconductors with Hc2 of the order of 25–30 kg at 1.1 K4. In 2008, a hydride, SiH4, revealed the metallic characteristic at 50 GPa and superconductivity of Tc ≈ 17 K at 100 GPa5.

From 2005, the high Tc was observed at 203 K and 150 GPa for H3S6, at 250 ~ 260 K and 180–200 GPa for LaH107, at 287 K and 274 GPa for a H–S–C compound8, and over onset 500 K for a LaH10 superhydride9. The first-principle calculations revealed a large density of states at the Fermi energy10,11. The isotope shifts of α = 0.50 ~ 0.35 (Tc Mα) measured for D2S6, α = 0.465 calculated by the first-principle approximation for LaD1012, and α = 0.4 experimentally evaluated for YD613, suggested that the electron–phonon interaction such as the BCS (Bardeen–Cooper–Schrieffer) s-wave superconductor6,12 is the pairing mechanism of superconductivity.

A particular feature of hydrides is a Tc divergence observed above a transition pressure, Ptransition, which leads to room-temperature superconductivity8,14,15, as shown in Fig. 1a. The Tc rise with the applied pressure is gradual below Ptransition and sharp over Ptransition. The gradual Tc rise is attributed to the small increase of the metal phase in the coexistence state of metal and insulator phases, while the sharp Tc rise results from the nearly single metal phase formed by the first-order insulator–metal transition (IMT)16,17; this is due to the percolation phenomenon. The IMT is not accompanied by any structural phase transition6,18. The IMT-percolation layout is shown in Fig. 1, which indicates that hydrides are the first-order IMT material undergoing percolation with increasing doping (or band filling), such as VO2 with inhomogeneity in the IMT process. This process implies hydrides are correlated materials. The first-order phenomenon has also been previously reported19.

Figure 1
figure 1

(a) Experimental data for the room-temperature Tc as a function of applied pressure8, which shows the Tc divergence over Ptransition = 220 GPa. The data were extracted in the paper8. The insulator–metal transition undergoes the first-order percolation (i.e., an increase in band filling) with increasing pressure (Inset). (b) A comparison of the weak coupling Tc (empty diamond) and the generalized Tc (filled square) in BCS theory. The coupling constant, b = 2Δ/kBTc, (blue ball) between the generalized energy gap and the generalized Tc has no restriction on the magnitude of λBCS. The Tc rapidly decreases below z = 3, which indicates that the generalized Tc does not explain the high Tc.

Regarding the room-temperature Tc, it may not be explained by the weak coupling BCS Tc with the electron–phonon coupling constant, λ ≤ 0.435, which describes the low-Tc superconductivity20. As an alternative, the strong-coupling McMillan Tc21 and the Allen-Dynes Tc22 without a restriction of the magnitude of λ have been suggested, although a max λMigdal N(0)VMigdal ≤ 1.5 has been given19. They are based on the Eliashberg formalism utilizing the increase in the Cooper-pair potential VMigdal with strong coupling23 and not the density of states N(0), the screened Coulomb repulsive potential μ, and the double potential well structure. μ depends on the number of carriers and is smaller in magnitude than the on-site short range repulsive Coulomb interaction, U. However, in the case of hydrides with a high Debye energy (ћω), due to the increase in the retarded Coulomb pseudo-potential, μ* = μ/(1 + μln(EF/ћω)) derived in conditions of λ << 1 and μ << 124, caused by a large deviation of ln(EF/ћω) > 1 in μ*, the exponential parts in the McMillan Tc and the Allen-Dynes Tc become much smaller than that obtained in BCS theory (see “Methods”). Although Allen-Dynes Tc, with ћωlog/1.2, an average of the phonon energy, different from ћω/1.45 as the prefactor of the McMillan Tc, is accurate at a small μ* value25, the Tc declines. This is due to the decrease in the exponential part in the Tc formula which is attributed to an increased value of μ* caused by a large Debye energy (see “Methods”)26. A comparison of the BCS Tc and the Tcs based on the Eliashberg formalism is shown25. Furthermore, an Tc exp [− 1/(λ-μ*)] derived in λ << 1 and μ* << 1 on the basis of Elisahberg formalism24 does not rise to room temperature, because λ′ = λ − μ* decreases with increasing μ* for hydrides. Therefore, the Tcs do not reach room temperature.

Subsequently, Migdal’s theory23 revealed that the increase in λMigdal, as strong coupling, results in the decrease in sound velocity proportional to the Debye energy, leading to the decrease in Tc. This finding indicates that a strong coupled model cannot explain the high Tc. Moreover, an exceedingly large λ = 6.2 was evaluated from experimental values using the McMillan Tc for YH627, which is much larger than the calculated value (λ = 1.71 ~ 2.24)13. The Eliashberg formalism does not fit the isotope effect11. Bogoliubov calculated the electron–phonon interaction by introducing the screened Coulomb repulsive interaction between electrons28, concluding that the screened Coulomb interaction plays little role in inducing superconductivity because the magnitude of the electron–phonon interaction is largely reduced by the Coulomb interaction. Thus, no theory is available to explain the high Tc. To enhance the Tc, the magnitude of density of states N(0) rather than the electron–phonon interaction should be increased. A BCS-based Tc that uses large N(0) as a function of band filling is needed.

In this report, we confirm the rise in Tc to room temperature by demonstrating the Tc divergence over Transition using a proposed BCS theory supported by the Brinkman-Rice picture29, with the diverging effective mass contributing to the density of states for a strongly correlated metal with U/Uc = κBR ≈ 1 (≠ 1). We reveal a fundamental cause of the electron–phonon interaction for superconductivity. The cause has remained obscure since the discovery of Onnes’s superconductivity in 1911, despite the development of BCS theory.

Derivations of superconducting-T c formulas

Generalized energy gap and T c in BCS theory

To overcome the weak coupling limitation of λ ≤ 0.435 in BCS theory, the energy gap of the Cooper pair and Tc need to be generalized. We find a generalized energy gap of the Cooper pair, a generalized Tc, and a generalized coupling constant between the energy gap and Tc without any restrictions in BCS theory. The energy gap, εg = Δ, of Eq. (2.40) in BCS theory20 is derived using sinh(x) = (ex − ex)/2 as follows:

$$\Delta = \frac{\hbar \omega }{{{\text{sinh}}\left[ {\frac{1}{{\lambda_{BCS} }}} \right]}} = \frac{{2\hbar \omega {\text{exp}}\left[ { - \frac{1}{{\lambda_{BCS} }}} \right]}}{{1 - {\text{exp}}\left[ { - \frac{2}{{\lambda_{BCS} }}} \right]}},$$
(1)

where ћω is the Debye’s phonon vibration energy, λBCS = N(0)Ve-ph is the electron–phonon coupling constant when the electron correlation is not considered, N(0) is the density of Bloch states of one spin per unit energy at the Fermi surface EF, and Ve-ph is a constant matrix element of the electron–phonon pair energy. Equation (1), satisfied with λBCS ≠ ∞, has a divergence in the denominator and has no restrictions on the magnitude of λBCS. In the case of λBCS ≤ 0.435, (which is the weak coupling limit confirmed by this author), Eq. (1) is reduced to the famous BCS energy gap, 2ћωexp(− 1/λBCS), by disregarding the extremely small value of exp(− 2/λBCS). At λBCS = 0.435, Δ/2ћω ≈ 0.1 is in the weak coupling limit of BCS theory. At λBCS > 0.435, the divergence of [1 − exp(− 2/λBCS)]−1 contributes to the enhancement of the energy gap. The derivation of Eq. (1) is given in the Supplementary Information.

As for superconducting Tc, the Tc equation of Eq. (3.28) in BCS theory20 is generalized without an approximation of a condition, Tc << ћω/kB = ΘD, and any restriction on λBCS, calculated as

$$T_{c} = C\left( z \right)\Theta_{D} exp\left[ { - \frac{\coth \left( z \right)}{{\lambda_{BCS} }}} \right],$$
(2)
$$\approx { }1.13\Theta_{D} exp\left[ { - \frac{\coth \left( z \right)}{{\lambda_{BCS} }}} \right],$$
(3)

where z = ΘD/2Tc is given, and \(C\left( z \right) \equiv \frac{1}{2}exp\left[ { - coth\left( z \right)\int_{0}^{z} {(ln\left( z \right)/cosh^{2} z)} dz} \right]\) is defined30. Here, to be the maximum Tc in Eq. (2), z should be ∞ in the function of C(z), after which coth(z) = 1 and max \(C\left( z \right) \equiv \frac{1}{2}exp\left[ { - \int_{0}^{\infty } {({\text{ln}}\left( z \right)/cosh^{2} z)} dz} \right] = \left( {\frac{{2e^{\gamma } }}{\pi }} \right)\approx 1.13\) are obtained, where γ ≈ 0.577 is the Euler constant. The derivation of Eq. (2) is given in the Supplementary Information. The Tc decreases with a decreasing z below z = 3, as shown in Fig. 1b. This phenomenon deviates from the limitation of the weak coupling BCS theory in which Tc is defined as over z = 3.

Moreover, the relation between the generalized energy gap Δ in Eq. (1) and the generalized Tc in Eq. (3) is given as

$$b = \frac{2\Delta \left( 0 \right)}{{k_{B} T_{c} }} = 3.54\frac{{exp\left( {\frac{1}{{\lambda_{BCS} }}\left[ {\coth \left( z \right) - 1} \right]} \right)}}{{1 - exp\left( { - \frac{2}{{\lambda_{BCS} }}} \right)}}.$$
(4)

The coupling constant, b, rapidly increases below z = 3 irrespective of a value of λBCS, as shown in Fig. 1(b), and it also increases over λBCS ≈ 0.435.

Superconducting T c driven by electron correlation

High-Tc superconductors with z < 3 have the Tc enhancement. In contrast, the Tc in Eq. (3) decreases, as shown in Fig. 1b. This means that Eq. (3) does not account for the increased Tc. Thus, to raise Tc, as a new concept, we assume the existence of the on-site Coulomb repulsive interaction (or correlation), U, between free electrons at the Fermi surface in a strongly correlated metal with U/Uc = κBR ≈ 1 (≠ 1) where Uc is a critical Coulomb interaction. The assumption is based on the first-principle calculations10,11, the divergence of the effective mass near the optimal doping31,31,33, and a suggestion that the strong correlation needs to be introduced34. The mass of carriers (quasiparticles) in the correlated metal is much heavier than that in the metal of BCS theory. As a result, the kinetic energy, εk, of the carriers, as expressed as εk = εBCS(1 − (U/Uc)2)2 with the effective mass of carriers m* = m/(1 − (U/Uc)2), is reduced with increasing U29. The kinetic energy does not contribute to the electron–phonon interaction35. Although εBCS is replaced by εk, the Hamiltonian and the Tc-formula form in BCS theory are not changed35. The BCS Tc equation was also solved by the Green function method36. The effect of the heavy mass of the carriers is independently compensated in the density of states for the Tc formula. Additionally, the inhomogeneity effect intrinsically appearing in the strongly correlated materials needs to be considered, which has been previously developed32,33.

Then, Eq. (3) is newly defined as follows;

$$T_{c,BR - BCS} \approx 1.13\Theta_{D}^{*} exp\left[ { - \frac{\coth \left( z \right)}{{\lambda_{BCS}^{*} }}} \right],$$
(5)
$$= 1.13\rho^{\frac{1}{3}} \Theta_{D} exp\left[ { - \frac{{{\text{coth}}\left( z \right)}}{{\left( {\frac{{\rho^{\frac{1}{3}} }}{{1 - \kappa_{BR}^{2} \rho^{4} }}} \right)\lambda_{BCS} }}} \right],$$
(6)

when ρ ≈ 1 from Eq. (6),

$$= 1.13\Theta_{D} exp\left[ { - \frac{{{\text{coth}}\left( z \right)}}{{\left( {\frac{1}{{1 - \kappa_{BR}^{2} }}} \right)\lambda_{BCS} }}} \right],$$
(7)

when coth(z) = 1 over z = 3 from Eq. (6),

$$= 1.13\rho^{\frac{1}{3}} \Theta_{D} exp\left[ { - \frac{1}{{\left( {\frac{{\rho^{\frac{1}{3}} }}{{1 - \kappa_{BR}^{2} \rho^{4} }}} \right)\lambda_{BCS} }}} \right],$$
(8)

where ΘD* = ρ1/3ΘD is an effective Debye temperature, λ* BCS is an effective coupling constant, and A N(0)*/N(0) = ρ1/3/(1 − κBR2ρ4) is a ratio of an effective 3D-density of states, N(0)* m*n1/3, at EF. In the two dimensional case, N(0)* m* is given. The λBCS is a constant, which is indefinite and must be extremely small. An effective mass of quasiparticles is given as m*/m ≡ 1/(1 − (U/Uc)2) = 1/(1 − ρ4) from U/Uc = κBRρ2 and, the correlation strength, 0 < κBR < 1 and, here, κBR≈1 (or 0.999…, not one)29,32,33 (Fig. 2a). A carrier density at EF, n = ρntot, is the extent of the metal region, 0 < ρ = n/ntot < 1 is the band-filling factor (or the normalized carrier density), and ntot is the number of all atoms in the measurement region32,33. ρ can be obtained from the Hall-effect experiment or the integral of the optical conductivity. ρ1/3 in ΘD* comes from the number of phonons in the phonon energy of lattices in the superconducting region (or metal phase over Tc) (inset in Fig. 2a). m* = m/(1 − ρ4) is obtained by applying an effective Coulomb energy, U/Uc = κBRρ2 and κBR 1, deduced in an inhomogeneous system to the Brinkman-Rice(BR) picture explaining the correlation effect in correlated metals formed by the impurity-driven IMT37,37,39, which is an average effect (or measurement effect) of the true effective mass, m* = m/(1 − κBR 2) at ρ = 132,33. The λBCS dependence of Tc,BR-BCS is shown in Fig. 2b. A large Tc change occurs in a small ρ variation near the half-filling ρ ≈ 1, confirming the presence of a divergence in the Tc formula. Moreover, when the λBCS value is slightly changed, ρ also varies. At a constant Tc, as λBCS increases, ρ decreases, but λ* does not change. Moreover, the physical meaning of the Tc,BR-BCS of Eq. (6) indicates an experimentally measured local Tc in the measurement region, which is an average (measurement effect) of the large intrinsic true Tc of Eq. (7) expressed by the true effective mass, m* = m/(1 − κBR 2), at ρ ≈ 1 in the BR picture29 (see Supplementary Information). The intrinsic true Tc of Eq. (7) is given as a function of κBR by applying ρ ≈ 1 into Eq. (6), which has a large diverging value near κBR = 1. The true Tc is constant determined at a given κBR 1 (≠ 1). The observed energy gap is obtained by replacing ћω and λBCS in Eq. (1) with kBΘD* and λ*, respectively. The coupling constant, b, is determined by substituting λBCS in Eq. (4) with λ*. Moreover, in the case of over z = 3, coth(z) in Eqs. (2) ~ (7) can be replaced with one and Eq. (8) becomes a BR-BCS Tc.

Figure 2
figure 2

(a) A divergence of an effective electron–phonon-coupling constant, λ* = BCS with λBCS = 0.03, is shown as a function of band-filling ρ, where A = N(0)*/N(0) = ρ(1/3)/(1 − κBR2ρ4) at κBR 1 is a ratio of the 3D density of states, and N(0) is the 3D density of states. The inset displays the divergences of the effective mass, m*/m = 1/(1 − ρ4)32,33, and the ratio A. In the inset, the layout of the inhomogeneous mixed phase with a correlated metal (κBR U/Uc ≈ 1 (≠ 1)) and insulator phases in the measurement region is also depicted. (b) The λBCS dependence of the BR-BCS Tc is shown. Here, the ΘD = 1250 K in Eq. (6) was used. As λBCS increases, Tc increases at a constant ρ. At a constant Tc, as λBCS increases, ρ decreases but λ* does not change.

Furthermore, we briefly note the physical meaning of ρ. For instance, it means that, in the case of ρ = 1, the whole measurement region is filled with a correlated metal of one electron per atom in real space, (inset in Fig. 2a), and the band is half-filled in k-space. In the case of ρ = 0.5, 50% of the measurement region is the metal in real space. Moreover, a condition of ρ = 1 is not defined due to the inability of U/Uc = 1 at m*/m = 1/(1 − (U/Uc)2) in the BR picture29. That is, neither the point of ρ = 1 nor half filling is attainable. This indicates that the correlated material is intrinsically inhomogeneous, which is the characteristic of the correlated material.

Results and discussions

In the superconducting state, the electron-phonon interaction, Ve-ph, forming the Cooper pair (pairing in k-space, time-reversed states) in BCS theory is fixed as a constant in real space and k space. This indicates the Cooper pair is a pair in real space (so called bipolaron), such as the pair potential Δ(r) proportional to Ve-ph = − V(r1,r2)δ(r1r2) suggested in the Bogolubov–de Genes (BdG) theory28,40,,40,41. The BdG theory derives the BCS formula for superconductors not only without impurities explained by BCS theory but also with nonmagnetic impurities both making a boundary between metal and nonmetal and not suppressing the superconducting gap42; this is an extension of the BCS theory. For a logical deduction of the constant, we consider an intersite charge-density-wave (CDW) potential as an electron-phonon interaction, VCDW = − (g2/2Mω2)δq2, such as the CDW with a charge disproportionation between nearest neighbor sites, δqδ(qiqj) = 2e, of BaBiO3 with the set Bi3+(6s2, the two electrons form bipolaron as a real-space pair) and Bi5+(6s0)43,44 (necessarily see “Methods”); the VCDW has an immobile bipolaron in real-space, thus indicating a set of both a paired occupied state (bipolaron) with two electrons on a site and an unoccupied state without electron at the nearest neighbor site. A range of the intersite CDW potential that reaches out in real space is within two lattice constants of 6~10 Å when the lattice constant in a metal is considered 4 ± 1 Å. Experimental evidence of the CDW in oxide superconductors is a distortion of octahedral structure observed just below Tc45,46 and discontinuity27 of the bulk modulus at Tc. For superconductivity, when the CDW potential is introduced, the on-site critical Coulomb energy Uc in the BR picture should be present at the bipolaron, then, as a nonlocal potential, Ve-ph = VCDW + Uc < 0 is considered a constant, because VCDW and Uc are determined as fixed values in a crystal. Since Uc is very large and constant, Ve-ph becomes extremely small or can approach but not reach zero; this explains why λBCS = N(0)Ve-ph should be small; further, N(0) is also small in an uncorrelated metal47 (see “Methods”). Then, the bipolaron can tunnel through the CDW potential to the next site; the supercurrent flows, which indicates the bipolaron has changed into the mobile Cooper pair in k-space (so called the mobile bipolaron) due to the Uc. Moreover, in the case of a strong coupling with a large Ve-ph, the Cooper pair can be trapped. Thus, we assert that Uc leads to superconductivity and that, although λ* in Eq. (6) is large (over one) (see Ti-2223 and Hg-1223 in Table 1), Tc of Eq. (6) is into weak coupling due to small Ve-ph in λBCS (Table 1).

Table 1 When experimental data in Fig. 1 are confirmed by Eq. (6), the obtained parameters are evaluated by the following formulas; m*/m 1/(1 − (U/Uc)2) = 1/(1 − κBR2ρ4) 1/(1 − ρ4) at κBR 1 (≠ 1), A = N(0)*/N(0) = ρ1/3/(1 − ρ4), and λ* = BCS, ΘD* = ρ1/3ΘD.

Subsequently, the coherence length was known as approximately ξ0 ≈ 5 Å34, within the range of two-lattice constant. The radius of the Cooper pair in real space48 was given as rCooper pair = πξ0. The coherence length, utilizing both the pair potential Δ(r) = Δ(0) at r = 0 calculated from the generalized BdG theory and the effective mass m*, was given as \(\xi_{0} = \frac{{\hbar {\text{v}}_{{\text{F}}} }}{\pi \Delta \left( 0 \right)} = \left( {\frac{\hbar }{{\pi {\Delta }\left( 0 \right)}}} \right)\sqrt {\frac{{2E_{F} }}{{m^{*} }}} ,\) where Δ(0) = 0.2ħωD and ξ0 = 0.2a for a nano crystal of a size of a = 15 nm was evaluated49. Moreover, Deloof et al.49 stated that the computational effect is reduced by increasing the effective mass and the coupling constant by decreasing the sample size. This author, according to the concept described here, adds that the large effective mass coming from the on-site Coulomb U can reduce the coherence length to a short range of two-lattice constant. A model of superconductivity based on the CDW has been reported44.

We apply the Tc of Eq. (6) to the experimental data for Tc with a transition pressure8, using ΘD  1250 K in a hydride mentioned by Ashcroft1. Note that the ΘD is not an accurate value because it is not yet known. The ΘD is used to check whether the Tc of Eq. (6) can rise to room temperature or not. The Tc values in Eq. (6) seem to rise to room temperature, as shown in Fig. 3. A relation of P vs. ρ is given in the caption of Fig. 3. The obtained parameters are given in Table 1. The obtained λ*s are over 0.435, the weak coupling limit of BCS theory. When precisely calculated ΘDs for the hydrides of H3S, D3S, LaH10, and LaHx are used50, the λ*s are also more than 0.435 and less than one (Table 1). We assert that the metallization is accelerated with increasing pressure, which is regarded as the increase in ρ. As evidence of the increased metallization induced by the first-order IMT, a jump in ρ is observed, as shown in Fig. 3. Furthermore, although λ*s are over one for Ti-2223 and Hg-1223 in Table 1, the large λ*s are caused by the large effective mass (large density of states) and not a large potential Ve-ph, such as the strong coupling potential VMigdal used in the Eliashberg formalism. Moreover, in Table 1, λ* = 0.384 for Pb, known as strong coupling of λ* = 1.1221 and 1.5522, is less than λ* = 0.435 of the weak coupling limit in BCS theory.

Figure 3
figure 3

The BR-BCS Tc of Eq. (6) and data in Fig. 1a are drawn together. The Tc calculations cannot be correct, because the Debye temperature, ΘD, is not correct; here ΘD = 1250 K was predicted in a hydride1, which indicates that Eq. (6) approaches the room-temperature Tc. The jump in ρ is observed as evidence of the first-order IMT. The detailed information is provided in Table 1. At line 1 over Ptransition ≈ 220 GPa, the relation between ρ and pressure P is P = 11,759.62ρ − 11,359.59, where the slope has a standard error of 747.35 and the standard error of the intercept is 737.64. At line 2 below Ptransition, the relation between ρ and pressure is given as P = 25,230.15ρ − 24,644.25, where the slope has a standard error of 4660.97 and the standard error of the intercept is 4588.50. The slope of line 1 is much larger than that of line 2, revealing the diverging behavior.

We briefly discuss a process of the IMT and a change in the correlation strength under high pressure. Compound materials are necessarily inhomogeneous and have an impurity level reflecting the semiconducting behavior. When pressure, temperature, strain, and chemical doping, among other energies are applied to the materials, the Mott-indirect IMT occurs by excitation of the impurity bound charges37,37,39. In the underdoped region, as the pressure increases, the extent of the correlated-metal region, ρ, increases due to the indirect IMT (percolation). Therefore, in some materials, at low temperatures, superconductivity appears. Decreasing the temperature reduces the size of the unit volume of the correlated metal (i.e., contraction of the unit volume), which causes an increase in the correlation strength. Additionally, applying pressure to the correlated materials leads to metallization as well as contraction of the unit volume, resulting in both an enhanced correlation and an increase of ρ. Thus, the density of states as a function of the effective mass diverges near ρ = 1 due to strong correlation of a constant value of κBRU/Uc ≈ 1 (not one), as shown in Fig. 2a. Thus, the Tc in Eq. (6) rapidly increases, which is the Tc divergence, as shown in Fig. 2b.

Furthermore, in the BCS-based mechanism for all kinds of superconductors, when the correlation effect in the density of states is introduced, the coupling constant, λBCS, should be replaced with λ* = AλBCS including the correlation effect. When λBCS < 0.1 with a small value47 (see “Methods”), instead of λ*, is applied to Eq. (5), Tc is not obtained; this is a weak point of BCS theory. This finding indicates that superconductivity does not occur without correlation; this is a mathematical discovery. Until now, to explain low-temperature superconductivity, a value near λBCS = 0.20 ~ 0.30 has been used, which should really be regarded as λ*. Moreover, the element superconductors explained by BCS theory should be regarded as correlated metals which are different from pure metals such as Au, Ag, or Cu that do not show superconductivity. The metallization in the element superconductors, including a non-metallic phase of few concentrations considered as impurity, is induced by the impurity-driven indirect IMT. This phenomenon is understood by observing the rise in Tc when pressure is applied to the element superconductors51,52, because the pressure effect does not appear in the pure metal crystals. Additionally, Eq. (6) can describe the high Tc of the cuprate superconductors. The λ* values obtained for important cuprate superconductors are given in Table 1. The energy gaps are slightly less than those we observed in the present analysis, which may be attributed to a smaller ΘD. We suspect that the observed ΘD was averaged to the multi-layered and inhomogeneous cuprate system, not measured on only the CuO2-layered plane. Accordingly, we assert that the superconductivity for all kinds of superconductors is caused by a change in the electron correlation that occurs due to the volume contraction induced by strong pressure or low temperature; this indicates that U in the correlated metal of the normal state can change to Uc of the condensed superconducting gapped state, which leads to the electron–phonon interaction at Tc.

Conclusion

The Tc,BR-BCS with the electron correlation of Eq. (6) accounts for the high Tc. It can be applied to all kinds of superconductors, such as element superconductors, compound superconductors, cuprate superconductors, and hydride superconductors, among others. The diverging Tc measured in the hydrides8 is responsible for the pressure-driven first-order IMT. Superconductivity can be attributed to the transition of the Bose–Einstein condensation from U to Uc, which derives from the volume contraction by applied pressure or low temperature.

Methods

Evaluation of the strong-coupled-McMillan T c

μ* μ/(1 + μln(EF/ћω)) should be satisfied with μ* << 121,24. μ* = (1 − 2α)0.5/ln(ΘD/1.45Tc) at λ <  < 1 was obtained from neglecting ‘strong-coupling’ correction term21. For D3S, α = 0.50 ~ 0.35 (Isotope effect6), ΘD = 869 K, and Tc = 155 K were determined50. For α ≈ 0.4658, μ* = 0.196 and for α ≈ 0.35, μ* = 0.405 are determined. In the case of max λ ≤ 1.519, for the McMillan Tc/(0.69ΘD) = exp(− [1.04(1 + λ)/(λ − μ*(1 + 0.62λ)]), Tc/(0.69ΘD) = 0.0985 ≈ 0.1 at both μ* = 0.196 and λ = 1.5 and Tc/(0.69ΘD) = 0.083 at both μ* = 0.405 and λ = 1.5 are obtained. The values of Tc/(0.69ΘD) ≈ 0.1 and 0.083 can correspond to (Tc/1.14ΘD) ≈ 0.1, the value of the weak coupling-limit of BCS theory. For instance, in the case of ΘD = 869 K and Tc = 155 K for D3S, from Tc/(0.69ΘD) ≈ 0.1, an obtained McMillan Tc ≈ 59.96 K is much smaller than Tc = 155 K. Thus, the McMillan Tc does not rise to the room-temperature Tc. Moreover, when the strong-coupling correction term of μ* = [(1 − 2α)(1 + λ)/(1–0.62λ)]0.5/ln(ΘD/1.45Tc) is utilized21, μ* = 0.223 for both α ≈ 0.46512, and λ = 1.5, and μ* = 0.461 for both α ≈ 0.35 and λ = 1.5 are calculated. Tc/(0.69ΘD) = 0.088 for μ* = 0.223 and 0.014 for μ* = 0.461 are obtained. For example, in the case of ΘD = 869 K and Tc = 155 K for D3S, from Tc/(0.69ΘD) ≈ 0.088, a McMillan Tc = 52.77 K, much smaller than Tc = 155 K, is determined. In particular, in the strong coupling, the Tc is smaller than that in the weak coupling. Thus, the McMillan Tc does not approach the room-temperature Tc.

Derivation of the charge density-wave potential, V CDW

For metal, we consider the breathing mode (harmonic oscillation) of an atom, then \(E_{Breath} = \frac{1}{2}kx^{2}\), where \(k = M\omega^{2}\), x is a small deviation from atomic position induced by the oscillation, M is a mass of the atom, and \(\omega {\text{ is atom}}^{^{\prime}} {\text{s oscillation frequency}}\). Next, for insulator, we consider the breathing mode distortion, \(E_{Breath - distortion} = g\delta qx\), where g is a proportional parameter, \(\delta q = q_{i} - q_{j}\) is a charge disproportionation between nearest neighbor sites. The total Energy, \(E_{CDW} = E_{Breath} + E_{Breath - distortion} = \frac{1}{2}kx^{2} + g\delta qx,\) is given. At a condition, \(\frac{{dE_{CDW} }}{dx} = 0,{ }x_{0} = - \frac{g\delta q}{k}\) is obtained. When x is replaced with \(x_{0}\) in \(E_{CDW}\), \(E_{CDW} = - \frac{{g^{2} \left( {\delta q} \right)^{2} }}{2k} = - \frac{{g^{2} \left( {\delta q} \right)^{2} }}{{2M\omega^{2} }},\) is obtained. On average of ECDW, \(< E_{CDW} > = - \frac{{ < g^{2} > \left( {\delta q} \right)^{2} }}{{2M < \omega^{2} > }},\) is given. When δq = 0, the electronic structure is one electron per atom of metal. In δq = 2e case, two electrons are occupied in a site and the nearest neighbor site is empty; this is the bipolaronic system. When δq = 1e, \(< E_{CDW} > = - \frac{{ < g^{2} > }}{{2M < \omega^{2} > }},\) is similar to λ/N(0) = \(\frac{2}{N\left( 0 \right)}\int {\frac{{d\omega \alpha^{2} \left( \omega \right)F\left( \omega \right)}}{\omega } = - \frac{{ < g^{2} > }}{{M < \omega^{2} > }}} ,\) in Eq. (23) (this is also CDW potential) in Ref.21 (MacMillan’s paper). When spin is considered, \(2 < E_{CDW} > = \frac{\lambda }{N\left( 0 \right)}\) is same. On the basis of this CDW logic, Eq. (23) in Ref.21 has an electronic structure in which one electron is occupied in a site and the nearest neighbor site is empty. Then, the number of electrons is half of total electrons in the system, which has a disagreement not satisfied with the metal condition (one electron per atom, that is, half filling) in the normal state (?); this is not bipolaron but just polaron. Finally, we assert that the electron–phonon interaction indicates the CDW interaction.

Approximate estimation of λBCS

The density of states of sulfur hydride was estimated to be 0.019 states/(spin-eV/Å3)25, when 2Δ ≈ 30.90 meV in Table 1 is approximately assumed as coupling potential Ve-ph, λBCS is given to be 0.587 × 10–3. When the density of states is calculated as 0.586 states/(spin-eV/Å3)50 obtained by assuming the standard BCS relation between energy gap and critical temperature, λBCS is determined to be 18 × 10–3. When, at most, 2Δ ≈ 60 meV is assumed, λBCS ≈ 36 × 10–3 can be evaluated, Thus, we assert λBCS is very small in an uncorrelated system.