HETERONUCLEAR SUPERCONDUCTORS WITH ELECTRONIC STRUCTURE INSTABILITY DRIVEN BY ELECTRON-PHONON COUPLING INTO

ANTIADIABATIC STATE

Field of the Invention

The invention relates generally to superconductors, in particular to heteronuclear solid state materials that are characterized by temperature dependent electronic structure instability that are superconductors at broken translation symmetry in stabilized intrinsic nonadiabatic or antiadiabatic electronic ground state. The present invention provides also methods for the design and synthesis of such superconductors.

Background of the Invention

Superconductivity is the state of zero resistance to electrical current in certain material when its temperature drops below a critical temperature T _c . In 1986, Bednorz and Muller discovered superconductivity in a lanthanum-based cuprate perovskite material at a T _c of 35 ⁰ K. In 1987, Chu and Wu found another material with a T _c of 92 ⁰ K. Much effort has been spent on high-Tc superconductors ever since their initial discovery. The main difficulty in the advancement in this research field is due to the lack of a theory that explains superconductivity at high temperatures. So far, the progress in the discovery of new high-T _c superconductive materials has been made mostly through empirical approaches.

A high T _c is thought be related to high value of electron-phonon ("EP") coupling constant (λ > 1) and high density of states nF at Fermi level. The standard, generally accepted EP-based theories of superconductivity, BCS or BCS-like theories, have been derived under the assumption of validity of the Migdal theorem and the Eliashberg restriction ("the ME approximation"). Migdal theorem is valid under the condition ωλ/Ep « 1, wherein ω is the phonon frequency, λ is the EP coupling constant, and Ep is the Fermi energy. The Eliashberg restriction restricts λ to be ≤ 1.

When expressed explicitly, BCS-like theories are valid only for adiabatic systems that obey the Born-Oppenheimer approximation ("BOA"): CO/E _F « 1. Only under these circumstances, the separation of electronic and nuclear motions is well justified, and the electrons and nuclei can be studied as two statistically-independent fields with mutual interaction that corrects the electronic energy and renormalizes phonon frequencies.

However, for high-T _c cuprates, fullerides, and MgB ₂ , it has been shown that Fermi energy of this group of materials falls on the same scale as the energy of the relevant optical phonon modes, E _F « ω. Problems with EP interactions within ME scenario gave rise to variety of non-phonon coupling models focusing on the role of electron correlations (EC). The underlying light motive behind the EC treatments has been to understand the phase diagram of high-T _c cuprates, i.e. the doping process. Recent results of angle-resolved photoemission spectroscopy ("ARPES") study of the group of high-T _c cuprates have brought experimental evidence that it is not doping, but an abrupt change (decrease) of the electron velocity near Fermi level (50-80 me V), that is the universal feature common to these compounds. Even more important in this respect is the formation of temperature- dependent giant kink close to Fermi level at and below T _c , which has been observed recently in Bi ₂ Sr ₂ Ca ₂ Cu ₃ O ₁₀ ("Bϊ2223") material. These ARPES results and the results of neutron scattering indicate that even for high-T _c cuprates, EP coupling has to be considered as a crucial element of microscopic mechanism of superconductivity state transition.

Studies of the dependence of band structure OfMgB ₂ on electron coupling to E _2g phonon mode (E _2g , 0.066 eV) have revealed even more important aspect crucial for microscopic theory of high-T _c superconductors. It has been shown that the vibration motion of boron atoms induces periodic fluctuation of the top (analytic critical point - see Figs.1 and 2) of one of σ band across Fermi level at the T point of the first Brillouin zone ("BZ"). Fermi level crossing occurs at nuclear displacement that is within the root-mean square ("rms") displacement of zero-point energy of this mode. It means that due to EP interactions, the E _F (chemical potential of σ band) is considerably reduced. From initial value E _F » 0.45 eV, it goes to 0 eV when the top of the band touches Fermi level. From the physical stand-point, it represents transition of the system from adiabatic to intrinsic nonadiabatic state, where ω > E _F , or even to antiadiabatic state, where rø » E _F . It has crucial theoretical impact. Under these circumstances, not only is ME approximation not valid (including impossibility to calculate nonadiabatic vertex corrections that represent off-diagonal corrections to adiabatic ground state), but the adiabatic BOA does not hold either.

The transition from adiabatic (ω « E _F ) to intrinsic nonadiabatic/antiadiabatic (ω » E _F ) state due to EP interactions seems to be the basic physical effect common for

superconductors. It means, however, breakdown of the adiabatic BOA, i.e. breakdown of the approximation that is the very basic starting point of many-body theory of solids. In this respect, it also means that different model treatments of superconductivity that are formulated in assumption of validity of the BOA (including BCS theory as well as models of strongly correlated electrons) are inadequate from the very beginning. The reason is evident, electronic Hamiltonian which is a basis of these models has been formulated in an implicit assumption of validity of the BOA that enables to separate electron - nuclear motion and then separately studied electronic system at "frozen" equilibrium nuclear configuration. On the level of the BOA, the motion of the electrons is a function of the instantaneous nuclear coordinates, but is not dependent on the instantaneous nuclear momenta. Situation for superconductors, as it shows ARPES results and study of band structure fluctuation, is substantially different. In this case, there is considerable reduction of electron kinetic energy that for intrinsic nonadiabatic state results even for dominance of nuclear dynamics (ω » E _F ) in some region of &-space. In this case, it is necessary to study electronic motion as explicitly dependent on the operators of instantaneous nuclear coordinates as well as on operators of instantaneous nuclear momenta. It is a new aspect for many-body theory of solids. There is, however, some experience with such a situation in theoretical molecular physics and it can be applied also for solids, particularly superconductors .

Summary of the Invention

The present invention encompasses heteronuclear solid state superconducting materials characterized by temperature-dependent electronic structure instability with respect to some of pertinent phonon modes that drive said materials from metal-like adiabatic electronic ground state with high-symmetry equilibrium nuclear geometry of corresponding crystal structure into superconducting intrinsic nonadiabatic or antiadiabatic geometrically degenerate electronic ground state that is stabilized at distorted nuclear geometry characteristic by fluxional configuration of the atoms in respective phonon modes, including, but not limited to, ZnB ₂ and LiB.

In one embodiment of the present invention, the heteronuclear solid state material is selected from stochiometric or nonstochiometric-electron or hole-doped solid state compounds, alloys, and composites.

The present invention encompasses methods of design and prepared by means of targeted synthesis, without necessity to perform numerous expensive trial syntheses, that is based on identification of phonon mode(s) that induce electronic structure instability which is characterized by fluctuation of electronic band structure. In one embodiment of the invention, the identification is by fluctuation of analytic critical point of some band across Fermi level due to electron-phonon coupling that is related to considerable increase of density of states at Fermi level, said fluctuation of electronic band structure being with respect to vibration displacements of the atom(s) for respective phonon mode(s) in never before synthesized or already synthesized solid state materials with particular composition and crystal structure.

The method can be any suitable method of electronic band structure calculation. Optionally, the method is based on Hartree-Fock approximation or Density functional theory approximation. The method of the present invention provide the necessary input parameters for determination of stabilization energy at transition from adiabatic to intrinsic nonadiabatic state for heteronuclear solid state superconducting materials and for determination of corrections to one-particle spectrum at said transition that is in a unique way related to thermodynamic properties, such as critical magnetic field and electronic specific heat, and to critical temperature T _c below which the heteronuclear solid state materials are superconductive. The heteronuclear solid state superconducting materials of this invention and the heteronuclear solid state superconducting materials designed by the methods of this invention can by prepared by any of standard methods of solid state synthesis or by any other method at proper synthetic conditions.

The present invention also encompasses technological applications of the heteronuclear solid state superconducting materials above and applications of said materials as a part of machines and devices

DESCRIPTION OF FIGURES Figure 1. Band structure OfMgB ₂ calculated at the equilibrium — undistorted geometry. The Fermi level - EF is indicated by the dashed line.

Figure 2. Band structure OfMgB ₂ calculated for the distorted geometry. The displacement of B atoms is | δf | = 0.005/atom (fraction unit) that correspond to 0.016 A ⁰ of B atom displacement out of equilibrium for stretching vibration (E _2g (a) phonon mode). At this displacement, the lower split-off σ band has just immersed below Ep _. The Fermi level - Ep is indicated by the dashed line.

Figure 3. Electronic ground state energy (per unit cell) OfMgB ₂ as the function of the displacement δf. The electronic energy of the undistorted structure is the reference value - 0 eV. The dependence is harmonic over the studied range of the displacements, and is identical for all studied types of the distortions. At the displacement | δf | = 0.005/atom, the lower split-off σ band has just immersed below E _F , and the ground state energy has been destabilized by 12 me V.

Figure 4. Equilibrium - undistorted high symmetry structure of B atoms in a-b plane and symmetry breaking by the in-plane, out of-phase displacements | δf | /atom. The motion of B1-B2 atoms in out-of phase positions along the perimeters of circles with radius δf centred at undistorted B-atom positions generates an infinite number of distorted structures. The figure indicates beside the undistorted-equilibrium structure also one of the distorted structures that corresponds to one of the possible nuclear configuration of the fluxional structure (bold line).

Figure 5. Dependence of the σι band density of states on the energy distance from the top of the band for MgB ₂ . Situation when the band-top touches Fermi level at T point.

Figure 6. The final — non-adiabatic density of the unoccupied states OfMgB ₂ at 0 K, according to Eq.(5). The uncorrected density of states is normalized to 1 and E _F = 0. The peak at 4 meV corresponds to density of states including non-adiabatic corrections of π band states and the peak at 7.6 meV corresponds to density of states including non- adiabatic corrections of σ ₂ band states. The density of the occupied states is the mirror picture with respect to E _F .

Figure 7. Part of the band structure OfYBa ₂ Cu ₃ O ₇ calculated at the experimental - undistorted equilibrium geometry. The fraction coordinates of the selected high symmetry points of the Brillouin zone are: G(≡T) (0,0,0), X(l/2,0,0), Y(0, 1/2,0), S(l/2, 1/2,0), Z(O ₅ 0, 1/2)

Figure 8. Band structure OfYBa ₂ Cu ₃ O ₇ at the coupling to A _g , B _2g and B _3g modes with displacements of apical 04 (Af ₀ = - 0.0027), planar O2 (δf _a = 0.0057) and 03 (δf _b = - 0.0057) atoms out of the equilibrium-experimental positions. At this distortion, there is a shift of the SP of one of the Cu(2)-O(2)/O(3) plane - derived ( d _χl _ ₂ - pσ ) band on the F-Y line at Y point, from below to above-Fermi level position.

Figure 9. Dependence of the d-pσ band density of states on the energy distance from the SP of the band for YBa ₂ Cu ₃ O ₇ . Situation when the SP touches Fermi level at Y point.

Figure 10. The final — non-adiabatic density of states OfYBa ₂ Cu ₃ O ₇ for the Ol - derived pσ band at Fermi level in F-Y direction (b) at k point where the band intersects Fermi level.

Figure 11. The final — non-adiabatic density of states OfYBa ₂ Cu ₃ O ₇ for the Ol- derived pσ band in F-X direction (a) at k point where the band intersects Fermi level.

Figure 12. Band structure OfZnB ₂ calculated at the equilibrium - undistorted geometry. The Fermi level - E _F is indicated by the dashed line. Figure 13. Band structure OfZnB ₂ calculated for the distorted geometry. The displacement of B atoms is | δf | = 0.009/atom (fraction unit) that correspond to 0.028 A ⁰ of B atom displacement out of equilibrium for stretching vibration (E _2g (a) phonon mode). At this displacement, the lower split-off σ band has just immersed below E _F . The Fermi level - E _F is indicated by the dashed line. Figure 14. Electronic ground state energy (per unit cell) OfZnB ₂ as the function of the displacement δf. The electronic energy of the undistorted structure is the reference value - 0 eV. The dependence is harmonic over the studied range of the displacements, and is identical for all studied types of the distortions. At the displacement | δf | = 0.009/atom, the lower split-off σ band has just immersed below E _F , and the ground state energy has been destabilized by 35 me V.

Figure 15. Dependence of the σpband density of states OfZnB ₂ on the energy distance from the top of the band. Situation when the band-top touches Fermi level at F point.

Figure 16. The final - non-adiabatic density of the unoccupied states OfZnB ₂ at 0 K, according to Eq.(5). The peak at 8.68 meV corresponds to density of states including non- adiabatic corrections of π band states and the peak at 14.92 meV corresponds to density of

states including non-adiabatic corrections of σ ₂ band states. The density of the occupied states is the mirror picture with respect to E _F -

Figure 17. Band structure of LiB calculated at the equilibrium - undistorted geometry. The Fermi level - Ep is indicated by the dashed line. Figure 18. Band structure of LiB calculated for the distorted geometry. The displacement of B atoms is | δf | = 0.007/atom (fraction unit) that correspond to 0.022 A° of B atoms displacement out of equilibrium for stretching vibration. At this displacement, the lower split-off σ band has just immersed below E _F . The Fermi level - E _F is indicated by the dashed line. Figure 19. The final — non-adiabatic density of the unoccupied states of LiB at 0 K, according to Eq.(5). The peak at 1.9 meV corresponds to density of states including non- adiabatic corrections of π band states and the peak at 3.4 me V corresponds to density of states including non-adiabatic corrections of σ ₂ band states. The density of the occupied states is the mirror picture with respect to E _F -

Detailed Description of the Invention

The present invention is based on the inventor's long term extensive study of electronic band structure of superconducting compounds and related nonsuperconducting analogues. The common feature of different classes of superconductors is the fluctuation of band structure related to EP coupling that induces electronic structure instability of the original adiabatic state of high-symmetry nuclear configuration. Under these circumstances, a system can be stabilized in the intrinsic nonadiabatic (ω > E _F ) or antiadiabatic state (ω » E _F ) that is related to the situation when the analytic critical point of the fluctuating band approach Fermi level on the energy distance less than ± ω and density of states at Fermi level is considerably increased. It also means that due to EP interactions that drive system from adiabatic to intrinsic nonadiabatic state, symmetry breaking occurs and system is stabilized in intrinsic nonadiabatic state at distorted geometry with respect to adiabatic equilibrium high symmetry nuclear configuration. Stabilization effect in the intrinsic nonadiabatic state is due to strong dependence of the electronic motion on the instantaneous nuclear kinetic energy, i.e. on the effect which is neglected on the adiabatic level within the BOA. Intrinsic nonadiabatic ground state at distorted nuclear configuration is geometrically degenerate with fluxional structure of

atom positions in the phonon modes that drive the system into this state. It has been found that while system remains in intrinsic nonadiabatic state, nonadiabatic polaron - renormalized phonon interactions are zero in well-defined k region of reciprocal lattice. This, along with geometric degeneracy of the intrinsic nonadiabatic electronic ground state, enables formation of mobile bipolarons that can move over lattice as supercarriers without dissipation. More over, it has been shown that due to EP interactions at transition to intrinsic nonadiabatic state, ^-dependent gap in one-electron spectrum is opened. Gap opening is related to a shift of the original adiabatic Hartree-Fock orbital energies and to the k- dependent change of density of states of particular band(s) at Fermi level. Corrected one-particle spectrum enables to derive thermodynamic properties that are in full agreement with corresponding thermodynamic properties of superconductors. System remains in intrinsic nonadiabatic state for temperatures T<T _C . Increase of temperature above T _c induces transition from intrinsic nonadiabatic state to non-superconducting adiabatic ground state at equilibrium high-symmetry nuclear configuration without the gap in one-particle spectrum.

From the disclosure presented above, for those skilled in the art is clear that it is possible not only to define but also to specify and derive the key physical parameters for the design of new superconducting heterostructures.

It is instructive to demonstrate it on the well known superconducting heteronuclear structures. a) Case OfMgB ₂ : superconductor, T ₀ » 39 K

The band structure OfMgB ₂ at equilibrium high-symmetry nuclear configuration of hexagonal omega phase (hP3, space group P6mmm, fractional coordinates of the unit cell atoms, Mg:0,0,0; Bl:l/3,2/3,l/2; B2:2/3,l/3,l/2, lattice constant: a = 3.0823, c = 3.5975) is in Fig.l. The band structure is of adiabatic metal character with relatively small Fermi energy of σ bands (E _F « 0.45 eV) and with very low density of states of these bands at Fermi level (calculated values are, n _Fσ ~ 0.02 in T-K direction and basically there is the same value in F-M direction for each band). Coupling to E _2g phonon mode (in-plane B-B stretching vibrations, ω » 0.066eV) induces splitting of σ bands degeneracy at T point. For B atoms displacement at about δf _a « 0.005, i.e. « 0.015 A°/B-atom out of equilibrium position, the top of lower σ band approaches Fermi level and crosses it. The band structure for this displacement is in Fig.2. Deformation potential of B-B stretching vibration is

harmonic over a wide range of B atoms displacements — Fig.3 presents the displacements up to δf _a * 0.008 (« 0.026 A°/B-atom). The root-mean square displacement (rms) calculated for this potential at zero-point energy is 0.039 A°/B-atom. It means, however, that at vibration motion at the displacement 0.015 A°/B-atom (that is much smaller than rms displacement already on the level of zero-point vibration energy) the system undergoes transition from adiabatic (Ep>ω) to intrinsic nonadiabatic state (Ep < ω) at the moment when the top of the σ band approaches Fermi level and crosses it. At this situation the BOA is not valid since the effective velocity of electrons related to the σ band (electrons in a-b plane - layer of B atoms) has been decreased and nuclear dynamics has to be considered — i.e. electronic motion is now dependent not only on the instantaneous nuclear positions but also on the instantaneous nuclear momenta. Standard many-body theories of solids do not solve this problem. Solution to this problem offers nonadiabatic molecular electron-vibration theory. According to it, for correction to the electronic ground state energy due to nonadiabatic EP coupling in quasi-momentum space representation holds,

(1)

Summation in (1) runs over all bands {R, S} and & points of 1. BZ of multi-band system, including intra-band contributions, i.e. R _t , Rr , k ≠ k', < ε _F ; ε _k , > ε _F .

At temperature 0 K (Fermi-Dirac occupation factors are: f _k =l,f _k '=0 ), for system in the intrinsic nonadiabatic state when top of the σ band is in the range of (± ω) eV from the Fermi level, calculated nonadiabatic correction δEfø to the electronic ground state is negative, ranging from -25 up to -98 meV/unit cell. System undergoes transition to intrinsic nonadoabatic state at vibration displacement 0.015 A°/B-atom that is related to an increase of the electronic energy by + 12 meV (Fig.3). However, nonadiabatic correction δE _(m) is in absolute value larger than increase in energy due distortion and consequently, system is stabilized at this distorted nuclear configuration. It can be easily shown that due to translation symmetry of the lattice, the electronic ground state corresponding to the intrinsic nonadiabatic state at distorted nuclear configuration is geometrically degenerate - there is an infinite number of energetically equivalent displacement vectors dβ of involved

couple of B nuclei (fluxional structure with atom positions on the perimeters of circles centered at the equilibrium B-atom positions with radius \d _B \ « 0,015 A ⁰ - see Fig 4).

Until the system remains in intrinsic nonadiabatic state, i.e. for temperatures T<T _C , cooperative revolving nuclear motion over perimeters of circles characteristic for this fluxional structure enables dissipationless motion of increased polarized intersite electronic charge density on the lattice scale. Temperature increase above critical value T ₀ induces transition from intrinsic nonadiabatic state to adiabatic state due to temperature dependence of AEfø (throughJermi-Dirac occupation factors f _k ,fr in (I)), and system is stabilized at undistorted equilibrium high-symmetry nuclear configuration. It has to be stressed that besides relatively strong EP coupling that drives system from adiabatic to intrinsic nonadiabatic state (calculated mean values of EP interaction matrix elements over energy interval ± ω at Fermi level are g _σσ « 0.7 eV and g _m * 0.25 eV), the crucial is considerable increase of the σ band density of states in the intrinsic nonadiabatic state when top of the band approaches Fermi level. In Fig.5 it can be seen that there is an increase from nF _σ « 0.02 that corresponds to density of states of σ band in the adiabatic state up to mean value n _Fσ « 0.5 in intrinsic nonadiabatic state when top of the band approaches Fermi level. From (1) is clear that such an increase of n _Fσ is crucial for stabilization of the system in intrinsic nonadiabatic state at distorted geometry.

At transition to intrinsic nonadiabatic state, not only ground state electronic energy is corrected (1), but corrected are also orbital energies \ε _k } - i.e. there is a change of the one- particle spectrum of the system, ε _k = ε _k + Aε _k . Shift of the orbital energies Aε^ in k point of band P, i.e. correction of dispersion in band P has the form,

m\>k _F ψ _v -ε _Vχ ) - ψω _k ,_ _k ,J sk<k _F \ε _v -ε _k ) - ψω _k _ _k .) (2)

for k ≤ k _F (3)

Replacement of discrete summation by integration introduces into above relations density of states n(ε _k ), is of crucial importance in relation to the fluctuating band, since at the moment when analytic critical point of the band approaches Fermi level (intrinsic nonadiabatic state), density of states at Fermi level is considerably increased as it has been shown above - Fig. 5. As the consequence of the upward-energy shift of the unoccupied states and downward-energy shift of the occupied states in the original quasi- continuum of states of the original adiabatic form, the temperature dependent gap in one- electron spectrum at Fermi level is opened;

The gap is opened close to the k point at the position where the band on adiabatic level has intersected Fermi level. The gap has character of indirect gap as it follows from relations (2, 3). This fact can be observed by tunnelling spectroscopy (at positive and negative biased voltage), or by ARPES (occupied states below Fermi level) and inverse ARPES spectra (unoccupied states above Fermi level) in the form of energy distance between peaks that are formed below and above Fermi level (spectral weight - density transfer) at decreasing temperature from above to below T _c . Since for corrected orbital energy holds, ε _k = ε _k ° + Aε _k , then for corrected density of states n{ε _k ) due to orbital energy shifts Aε _k , the following relation can be derived,

The quantity in relation (5) stands for uncorrected density of states of particular band (density of states on adiabatic level),

In the case OfMgB ₂ , the E _2g phonon mode mediates coupling between two σ bands (g _σσ « 0.7 eV) and weaker coupling is between the σ band with fluctuating band-top across Fermi level and π band (g _σπ « 0.25 eV). These interactions result in orbital energy shifts in σ and π bands at Fermi level (according to 2, 3). It opens two gaps in originally quasi-

continuum of states of σ and π bands at Fermi level. Calculated final density of states (5) for energy region above Fermi level is presented in Fig.6. Energy distance from the Fermi level of the peak at lower energy (4 me V) corresponds to the half-gap associated with the π band, and that of the second one at higher energy (7.6 me V) corresponds to the half-gap associated with second σ band (σ band with the top above Fermi level). Critical temperature T _c (above this temperature system losses superconducting properties, i.e. zero electric resistance and diamagnetism) is within the disclosed invention related to the temperature at which system that is at temperature 0 K stabilized in intrinsic nonadiabatic state at distorted nuclear configuration undergoes transition to adiabatic metal-like state at equilibrium undistorted high-symmetry nuclear configuration. It means, however, that at this temperature (and above it) the gap at Fermi level has to disappear (A(T _C ) = 0) and system remains at equilibrium high-symmetry nuclear configuration with quasi-continuum of states at Fermi level. At this conditions, for T _c can be derived from (4) the following expression,

T. -ψ O)

In the case OfMgB ₂ , the larger gap corresponds to the σ band and at temperature 0 K it has the value δ(0) = 15.2 me V. Critical temperature, according to (7), that corresponds to this gap is, T _c « 39.5 K. The value of T _c and density of states (Fig. 6) are in full agreement with experimental data as obtained, e.g. by tunnelling spectroscopy, for this superconductor.

It can be shown that electronic structure of nonsuperconducting diborides (e.g. AlB ₂ , ScB ₂ , YB ₂ , TiB ₂ , ZrB ₂ , HfB ₂ , NbB ₂ , TaB ₂ ,....), or isoelectronic compound LiBC, with the same crystal structure as MgB ₂ (hP3, space group P6mmm, omega phase) is stable at equilibrium high-symmetry nuclear configuration and coupling to pertinent phonon modes (e.g. E _2g ) do not drive these systems into intrinsic nonadiabatic state.

b) Case OfYBa ₂ Cu ₃ O ₇ : the first member of the family of high-T _c cuprate superconductors, T _c « 92 K

The YBa ₂ Cu ₃ O ₇ is of orthorhombic structure (space group Pmmm, oP14), with the fraction coordinates of the unit cell atoms: Cu(I) = (0,0,0); Cu(2) = (0,0, +0.355); Y =

(1/2,1/2,1/2), Ba = (1/2,1/2, ±0.186), 0(1) = (0,1/2,0), 0(2) = (1/2,0, ±0.380), 0(3) = (0,1/2, ±0.376), 0(4) = (0,0, ±0.156) and lattice constants a = 3.817 A ⁰ , b = 3.882 A ⁰ , c = 11.671 A ⁰ . The unit cell has 13 atoms as it corresponds to the formula unit with the chain oxygen 0(1) in b-direction and vacancy in a direction. The band structure OfYBa ₂ Cu ₃ O ₇ at equilibrium high-symmetry nuclear configuration is in Fig. 7. Important is the topology of the couple of Cu(2)-O(2)/O(3) planes - derived (d _χ2 _ ₂ -per) bands with the maximum in antibonding region at the S point. Both bands cross Fermi level, enter the bonding region on the S-X, respectively S-Y lines and intersect X, respectively Y points, well below the Fermi level. The chain oxygen O(l)-derived pσ band is going up from the Y point to S point where it reaches maximum, then continues with a little dispersion to X point , from this point starts to fall-down, intersects Fermi level and enters bonding region on the line X-F and second intersection is on the line Y-F. This band structure has, like band structure OfMgB ₂ , adiabatic metal-like character with low density of states at Fermi level. Density of states of the chain oxygen Ol -derived pσ band at k-point(s) where the band intersects Fermi level calculated from the band structure is for F-Y direction n _rγ — 0.04, and for direction F-X corresponding value is even smaller, n _τx = 0.03. Density of states of ( d _{2 2} - pσ ) bands in S-X and S-Y directions at Fermi level are basically of the same small values. Study of the influence of EP coupling on the band structure shows that combination of O4 (A _g phonon mode) , 02, O3 atoms vibration (B _2g and B _3g phonon modes, ω * 0.072 eV) shifts periodically up and down the saddle point (SP) of one of the Cu(2)-O(2)/O(3) plane - derived ( d _χl _ ₂ - pσ) band on the F-Y line at Y point across the

Fermi level. At the displacements of planar 02 by δf _a = 0.0057 (0,0217 A ⁰ ) and 03 by δf _b = - 0.0057 (-0.0221 A ⁰ ) and apical 04 by δf _c = - 0.0027 (-0.0315 A°), the SP of the (d _χ i_ 2 - per ) band at Y point has approached and crossed the Fermi level from below to above-Fermi level position - Fig. 8. Keeping fixed the displaced position of O4 with continuation in displacements of 03, 02 atoms shifts the SP back across the Fermi level to below-Fermi level position and the band structure of the topology of Fig.7 is recovered. The same fluctuation of the SP across Fermi level is reached when specified displacements of 02, O3 atoms are fixed and 04 is displaced. At the moment when the SP at Y point approaches Fermi level, i.e. at the specified vibration displacements, increase of the ground state electronic energy is + 170 meV/unit cell. At this situation, however, the BOA

is not valid; system is in the intrinsic nonadiabatic state. At the transition to intrinsic nonadiabatic state calculated mean values of EP interaction matrix elements over energy interval ± ω at Fermi level is g« 2.5 eV. The EP coupling mediates interactions between Cu2-O2/O3 (d _χ2 _ ₂ - per ) band and chain oxygen 0(1) - pσ band electrons. When the SP Of(^ _2- j - pσ) band approaches Fermi level, density of states of this band at Fermi level are increased considerably - Fig. 9. For energy distance up to ±ω from Fermi level, mean value of density of states of this band is nFdσ-p∞ 2. At this circumstances calculated nonadiabatic correction δE _(na) (1) to the electronic ground state is negative with mean value equal to, δE ₍ ,, _a) « - 204 eV/unit cell. The nonadiabatic correction AEfø is in absolute value larger than increase in energy due distortion and consequently, system is stabilized (— 34 me V) at the distorted nuclear configuration. Due to translation symmetry of the lattice, the electronic ground state corresponding to the intrinsic nonadiabatic state at distorted nuclear configuration is geometrically degenerate - there is an infinite number of energetically equivalent displacement vectors do? and do ₃ (fluxional structure with O2 and 03 atom positions on the perimeters of circles centered at the equilibrium positions of respective atoms, with radius J ₀₂ 1« 0.0217 A ⁰ and d _O3 « 0.0221 A ⁰ ). Until system remains in intrinsic nonadiabatic state, i.e. for temperatures T<T _C , cooperative revolving nuclear motion of 02 and 03 atoms over perimeters of circles characteristic for this fluxional structure enables dissipation-less motion of increased polarized intersite electronic charge density on the lattice scale. In the intrinsic nonadiabatic state at the distorted geometry, the gap in one-particle spectrum is opened in Ol-pσ band at Fermi level in b direction (F-Y line) and in a direction (F-X line). Calculated final density of states, according to (2, 3, 5), in particular directions are shown in Fig. 10 and Fig. 11. The larger gap is in b direction, Fig.lO, with calculated value δ _b (0) = 35.7 me V, while smaller gap is in the a direction, Fig.11 , with calculated value δ _b (0) = 24.2 me V. Calculated value, according to (4), of the critical temperature that corresponds to the larger gap is T _c = 92.8 K. The critical temperature corresponds to temperature at which the system undergoes transition from intrinsic nonadiabatic state (superconducting ground state) to adiabatic state that does not exhibit superconducting properties. The value of T _c and ratio of gaps in

a and b directions, « 0.68 for this superconductor.

It can be shown that electronic structure of nonsuperconducting YBa ₂ Cu ₃ O ₆ (chain oxygen-Ol deficient) with the same crystal structure as YBa ₂ Cu ₃ O ₇ (space group Pmmm, oP14) is stable at equilibrium high-symmetry nuclear configuration and coupling to pertinent phonon modes (A _g , B _2g and B _3g ) do not drive these systems to intrinsic nonadiabatic state.

Examples Example 1

The boride ZnB ₂ with hexagonal structure (hP3, space group P6mmm, omega phase) has been found by the method of present invention to be two-gap superconductor with T _c « 77.5 K, which is higher than the boiling point of nitrogen. Figure 12 presents the band structure OfZnB ₂ at optimized (equilibrium) high- symmetry geometry (lattice constant: a = 3.0725, c = 3.3209, fractional coordinates of the unit cell atoms, Zn: 0,0,0; Bl: 1/3,2/3, 1/2; B2:2/3,l/3,l/2). The band structure is of adiabatic metal character with relatively small Fermi energy of σ bands and with very low density of states of these bands at Fermi level (calculated values are, n _Fσ » 0.02 in F-K direction and basically there is the same value in F-M direction for each band and for π band it is nF _π « 0.017 in both direction). Coupling to E _2g phonon mode (in-plane B-B stretching vibrations, ω « 0.083eV) induces splitting of σ bands degeneracy at F point. For B atoms displacement at about δf _a « 0.009, i.e. » 0.028 A°/B-atom out of equilibrium position, the top of lower σ band approaches Fermi level and crosses it. The band structure for this displacement is in Fig.13. Deformation potential of B-B stretching vibration is harmonic over a wide range of B atoms displacements - Fig.14 presents the displacements up to δf _a » 0.018 (» 0.055 A°/B-atom). The root-mean square displacement (rms) calculated for this potential at zero-point energy is 0.036 A°/B-atom. It means, however, that at vibration motion at the displacement 0.028 A°/B-atom, the system undergoes transition from adiabatic (E _F >ω) to intrinsic nonadiabatic state (E _F < ω) at the moment when the top of the σ band approaches Fermi level and crosses it. Transition to intrinsic

nonadiabatic state is related to an increase of the electronic energy by + 35 meV (see Fig.14 at vibration displacement δf _a « 0.009, i.e. « 0.028 A°/B-atom). In the intrinsic nonadiabatic state when top of the band approaches Fermi level, there is considerable increase of the σ band density of states. In Fig.15 it can be seen that there is an increase from n _Fσ « 0.02 that corresponds to density of states of σ band in the adiabatic state up to mean value n _Fσ « 0.5 in intrinsic nonadiabatic state when top of the band is ±ω/2 from the Fermi level. There is also strong EP coupling that drives system from adiabatic to intrinsic nonadiabatic state (calculated mean values of EP interaction matrix elements over energy interval ± ω at Fermi level are g _σσ ∞ 1.34 eV and g _σπ * 0.61 eV). Under these circumstances, calculated nonadiabatic correction δE? \ (eq.l) to the electronic ground state is negative with mean value AEfø∞ - 75 meV/unit cell at temperature 0 K. Since nonadiabatic correction δEfø is in absolute value larger than increase in energy due distortion, system is stabilized (by about -40 meV/unit cell) at distorted nuclear configuration. It can be easily shown that due to translation symmetry of the lattice, the electronic ground state corresponding to the intrinsic nonadiabatic state at distorted nuclear configuration is geometrically degenerate - there is an infinite number of energetically equivalent displacement vectors dβ of involved couple of B nuclei (fluxional structure with atom positions on the perimeters of circles centered at the equilibrium B positions with radius 0,028 A°). Until system remains in intrinsic nonadiabatic state, i.e. for temperatures T<T _C , cooperative, revolving, nuclear motion over perimeters of circles characteristic for this fluxional structure enables dissipation-less motion of increased polarized intersite electronic charge density on the lattice scale. Temperature increase above critical value T _c induces transition from intrinsic nonadiabatic state to adiabatic state due to temperature dependence of ^E _(m) (through Fermi-Dirac occupation factors fu,fk- in (I)), and system is stabilized at undistorted equilibrium high-symmetry nuclear configuration.

Shift of the orbital energies As ₁ ^ (eqs. 2, 3) results in opening of gaps (eq.4) in one-particle spectrum of σ and π bands with the corresponding change of the density of states at Fermi level. In particular case of the ZnB ₂ , the half-gap of σ band is δσ/2 « 14.9 meV and that of π band is δπ/2 « 8.7 me V. Critical temperature, according to (7), that

corresponds to σ band gap is, T _c » 77.5 K. Calculated final density of states (5) for energy region above Fermi level is presented in Fig.16.

Example 2 The boride LiB with hexagonal, graphite-like structure and alternating honeycomb- like layers of boron and lithium atoms (hP4, space group P6 ₃ / mmc, fraction coordinates of the unit cell atoms, Lil:0,0,l/4; Li2: 1/3,2/3,1/4; Bl:0,0,3/4; B2:2/3, 1/3,3/4, optimized equilibrium lattice constant: a = 3.2692, c = 4.3695) has been found by the method of present invention to be superconducting with critical temperature T ₀ » 17 K in antiadiabatic state at distorted geometry. The band structure of LiB at equilibrium geometry is in Fig.17. It is of adiabatic metal-like character with relatively small Fermi energy of σ bands and with very low density of states of these bands at Fermi level (calculated values are, n _Fσ « 0.025 in F-K direction and basically there is the same value in F-M direction for each band and for π band it is nF _π « 0.02 in F-K direction). Coupling to B-atoms in-plane stretching vibration phonon mode induces splitting of σ bands degeneracy at F point. For B atoms displacement at about δf _a « 0.007, i.e. « 0.022 A ⁰ ZB- atom out of equilibrium position, the top of lower σ band approaches Fermi level and crosses it. The band structure for this displacement is in Fig.18. At the crossing, the EP coupling that drives system from adiabatic to intrinsic nonadiabatic state is calculated to be g _σσ * 0.44 eV and g _σπ « 0.1 eV and density of states of σ band increases to nF _σ * 0.3 at the moment when top of the band approaches Fermi level. Under these circumstances, the system is stabilized in antiadiabatic state at distorted geometry (approximately by -20 meV). Nonadiabatic corrections to single-particle spectrum open the gap in σ and π band at Fermi level. For temperature 0 K, the half-gap of σ band is δσ/2 « 3.4 me V and that of π band is δπ/2 « 1.9 me V. Critical temperature, according to (7), that corresponds to σ band gap is, T _c « 17.1 K. Calculated final density of states (5) for energy region above Fermi level is presented in Fig.19.