FIELD OF THE INVENTION

[0001] This invention is related to the field of high critical temperature Tc superconductivity and applications.

BACKGROUND OF THE INVENTION

[0002] A class of superconductors, henceforth referred to as high-Tc superconductors, was first discovered by Bednorz and Muller and disclosed in Possible High-Tc Superconductivity in the BaLaCuO System, Zeitschrift fr Phsik, B64, 189 (1986) . Since then several kinds of this class of superconductors of the cuprate type have been discovered. The Bi-based cuprate with critical temperature Tc of 105 K was disclosed in H. Maeda, et al., J. App. Phys. 27, L209 (1988), and in US Patent 7132288, Maeda et al. 1/1989, The Tl-based cuprate with Tc of 125 K was disclosed in Z.Z. Sheng and A.M. Hermann, Nature, 332, 55 (1988), and in US Patent 5082825 Hermann 8/1988. The Hg-based cuprate with Tc of 135 K was disclosed in A. Schilling, et al., Nature, 363, 56 (1993), and in L. Gao, et al., Phys. Rev. B, 50, 4260 (1994), and in US Patent 5578551 Chu et al. 11/1996. Up to now the highest critical temperature Tc is about 164 K for Hg-based cuprate under high pressure. Now a main aim of the field of superconductivity is to understand the mechanism of the high-Tc superconductivity and to find high-Tc superconductors with the critical temperature Tc > 300 K.

[0003] Recently a quantum gauge model of high-Tc superconductivity is presented by the inventor (c.f. Sze Kui Ng, Gauge model of high-Tc superconductivity, the 26th Conference on Low Temperature Physics, Beijing, August 2011, and Sze Kui Ng, arXiv:math/0004151v3) . This model unifies the electron-electron interaction and the electron-phonon interaction for the forming of Cooper pairs of electrons of superconductivity. In this gauge model, there is a seagull vertex term: - |]~pj; ^e o Z ^c Z A ² where the complex variable Z represents an electron ( Z ^c is the complex conjugate of Z ) , the real variable A represents the photon gauge field, and ^■ jj - _j ^ is a parameter where h is an energy parameter proportional to the Planck constant and the parameter k = k g T where k g is the Boltzmann constant and T is the absolute temperature, and e o is the (bare) electric charge. In this gauge model, while there is a term of order e o gives the usual repulsive Coulomb potential of order e between electrons, this seagull vertex term of order e gives a small attractive potential of order e between electrons. Since for a crystal material the repulsive Coulomb potential between electrons is balanced by the attractive potential from phonons (or protons) interacting with electrons, this small attractive potential can be used as an additional electron-electron interaction for the forming of Cooper pairs of electrons. By a renormalization group method, this gauge model gives a unified description of superconductivity and magnetism including antiferromagnetism, pseudogap phenomenon, paramagnetic Meissner effect, Type I and Type II supeconductivity and high-Tc superconductivity. In this gauge model, the doping mechanism of superconductivity is found. It is shown that the critical temperature Tc is related to the ionization energies of elements and can be computed by a formula of Tc. For the high-Tc superconductors such as La(2-x)Sr(x)Cu04, YBa2Cu307, and MgB2, the computational results of Tc agree with the experimental results.

[0004] In this gauge model of high-Tc superconductivity the inventor shows that the phase of

SUBSTITUTE SHEET (RULE 26) high-Tc superconductivity (and the conventional Type II superconductivity) appears at the phase line h ² = 3 k ² . It is shown that for a f ixed k the increasing of the doping parameter x of the cuprates such as La(2-x)Sr(x)Cu04 gives the increasing of h. Thus the phase plane ( k, h ) of phase diagram corresponds to the phase plane ( T, x ) of phase diagram of superconductors such as La(2-x)Sr(x)Cu04. It is shown that this increasing of h by the increasing of x gives the doping mechanism of high-Tc superconductivity.

[0005] In this gauge model of high-Tc superconductivity by using the renormalization group method we get a nontrivial fixed point (r, u) of the renormalization group equation. This fixed point is in terms of a parameter epsilon which is analogous to the epsilon parameter in the epsilon-expansion of the Ginzburg-Landau model of superconductivity in statistical physics. As analogous to the epsilon parameter of the GL model this parameter epsilon is related to the dimension of the space, denoted by d, such that d=3-epsilon is the (fractional) dimension of the space.

[0006] From this fixed point we have two cases. The Case 1) is with 0 < epsilon << 1 which gives the three dimensional (3D) Type I and Type II conventional superconductivity while the Case 2) is with epsilon = 1 which gives the quasi-2D high-Tc superconductivity. The formula of this f ixed point has a factor . From the expansion of this factor and from the property of the fixed point, the phase plane ( T, x ) of the phase diagram of high-Tc superconductivity is derived.

SUMMARY OF THE INVENTION

[0007] A method of composing superconductors with the critical temperature Tc > 300 K is disclosed. This method is from the abovementioned gauge model of high-Tc superconductivity wherein the doping mechanism of superconductivity is found. A class of superconductors composed by this method is the class of intermetallics with a hexagonal crystal structure consisting of three layers wherein the middle layer is as the conducting layer.

[0008] A kind of superconductors with hexagonal crystal structure composed by this method is the AlB2-type intermetallic superconductors which are obtained by doping the AlB2-type intermetallics such as A(l-x)Ca(x)Ga2 ( 0 < x < 1 ) wherein A=Sr, Ba.

[009] A kind of superconductors with hexagonal crystal structure composed by this method is the CaCu5-type intermetallic superconductors which are obtained by doping CaCu5-type intermetallics such as La(l-x)A(x)Cu5, Y(l-x)A(x)Cu5, Mm(l-x) A(x)Cu5, ( A=Ca, Sr; Mm denotes Mischmetal), LCu(5(l-x))Ni(5x) (L=La, Y, Mm; 0 <= x <= 1), and Sr(l-x)Ca(x)Cu5 ( 0 < x < 1 ), and La(l-x)Sr(x(l-y))Ca(xy)Cu5 ( 0 < x, y < 1 ), In particular the CaCu5-type intermetallics LaNi5 and MmNi5 are superconductors with critical temperature Tc > 300 K. It is shown that these CaCu5-type superconductors are with high critical current densities and thus are applicable for the transmission of electricity.

DESCRIPTION OF DRAWINGS

[0010] FIG.l shows the ( T, x) phase diagram of high-Tc superconductors such as La(2- x)Sr(x)Cu04. This phase diagram is derived from the abovementioned gauge model of high- Tc superconductivity and the well known experimental ( T, x) phase diagram of La(2- x)Sr(x)Cu04 in R.J. Cava, et al., Phys. Rev. Lett., 58, 408 (1987), and W.Y. Liang, et al., J. Phys. : Condens. Matter, 10, 11365 (1998) .

[0011] In this phase diagram the parameter k corresponds to T by the relation k = k g T. The line a denotes the phase line k ² = 3 h ² ( of the phase plane ( k, h) in the phase plane ( T, x) . The line b denotes the cross-over line k ² = h ² (of the phase plane ( k, h) in the phase plane ( T, x) . The line c denotes the phase line h ² = 3 k ² (of the phase plane ( k, h) of superconductivity in the phase plane ( T, x) . Then the line d denotes the cross-over line h ² = (2 + 35 ) ² k ² (of the phase plane ( k, h) in the phase plane ( T, x) . In this phase diagram the original phase line h ² = 3 k ² with the part denoted by the dash line is bifurcated into the phase line h ² = 3 k ² denoted by the line c in the phase plane ( T, x) .

[0012] The two lines a and c are the two basic phase lines for antiferromagnetism and superconductivity respectively. In this phase diagram we have two more phase lines b and d. The region between a and b is the region of spin density waves and region of insulator. The region between b and c is the region of charge density waves and region of semiconductivity. The region between a and c is usually called the region of pseudogap. The region between c and d is the region of paramagnetic Messiner effect and is usually called the region of non- Fermi liquid (We may also call this region as the extended region of pseudogap) . The right side of d is the region of normal metallic state. Then the bifurcation of the phase line c gives the region of high-Tc superconductivity. Then the left side of a is the region of antiferromagnetism (The region of antiferromagnetism is also obtained by the bifurcation of the phase line a where a part of this region of antiferromagnetism obtained by bifurcation is at the outside of this phase diagram in FIG.l) .

[0013] Then let us consider the special phase line e . This phase line e is the cross-over line from the phase of two dimensional phenomenon (i.e. the Case 2) with epsilon=l) to the phase of three dimensional phenomenon (i.e. the Case 1) with 0 < epsilon << 1 ) . Above this line is the phase of two dimensional (2D) phenomenon and below this line is the phase of three dimensional (3D) phenomenon. In the 2D phase the region in the bifurcation region of superconductivity is the region of high-Tc superconductivity and in the 3D phase the region in the bifurcation region of superconductivity is the region of conventional Type II superconductivity. In the 3D phase the region between the line c and the line d is the region of Type I superconductivity (In the 2D phase this region between the line c and the line d is the extended region of pseudogap) .

[0014] Thus this phase diagram in FIG.l is a complete phase diagram on magnetism and superconductivity that it includes the Type I superconductivity, the conventional Type II superconductivity and the high-Tc superconductivity. The existence of the 3D phase in this phase diagram is important for the 2D phase high-Tc superconductivity since the existence of the 3D Type II superconductivity gives the stable existence of the 2D high -Tc superconductivity. This existence of the 3D Type II superconductivity is as the reservior for the stable existence of the 2D high -Tc superconductivity.

[0015] FIG.2 shows a hexagonal unit cell of MgB2 of the AlB2-type. The B atoms are depicted by the black balls and the Mg atoms are depicted by the white balls. When Mg is replaced by Ca(x)Sr(l-x) and B is replaced by Ga, it also shows a unit cell of Ca(x)Sr(l-x)Ga2 of the AlB2-type. [0016] FIG.3 shows two layers of the unit cell of La(l-x)Ca(x)Cu5 of the CaCu5-type. This unit cell is with the hexagonal crystal structure of three layers. FIG.3a is the middle layer of the hexagonal crystal structure and FIG.3b is the lower or upper layer of the hexagonal crystal structure. The Cu atoms are depicted by the black balls and the La(Ca) atoms are depicted by the white balls.

DESCRIPTION OF PREFERRED EMBODIMENTS

[0017] In this gauge model of high-Tc superconductivity the doping mechanism of superconductivity is found. Also a method of computation of critical temperature Tc of superconductors is found. In particular the doping mechanism of superconductivity and the computation of the Tc of the intermetallic MgB2 (US Patent 6956011 Akimitsu et al., 10/2005) are given. This intermetallic MgB2 is of the hexagonal AlB2-type crystal structure. In this invention we extend the doping mechanism of superconductivity and the computation of Tc of MgB2 to other intermetallics. To this end let us first consider the intermetallic Mg(l-x)Be(x)B2 ( 0 <= x <= 1) . This intermetallic is of the AlB2-type. For the doping mechanism of superconductivity of this intermetallic, let us consider the following function:

f(x)=737.7(l-x)+899.5x (1) where 737.7 kJ/mol and 899.5 kJ/mol are approximately the first ionization energies of Mg and Be respectively. This function gives the relation that the increasing of x gives the increasing of h. We have 737.7 < 800.6 < 899.5 where 800.6 kJ/mol is the first ionization energy of B. Let us set the following relation (A main point is that 800.6-737.7 is small) :

f(x ₀ ) = 737.7 (1- x ₀ ) + 899.5 x ₀ = 800.6 (2) for some x o ( 0 <= x o <= 1) . When (2) holds (or approximately holds), the channel connecting the two states of first ionization energy of B and Mg is opened (We notice that the relation (2) gives the degeneration of electron states. The resulting degenerate state gives a Jahn- Teller electron-phonon interaction effect which is described as a channel opening. Let us call (2) and the resulting Jahn- Teller effect of degeneration as the degenerate state of channel opening) . This channel opening gives a freedom of electrons of B with a direction orthogonal to the B plane. By this freedom of electric current, the electrons of a cluster of B atoms in a unit cell can be coupled to the electrons of cluster of B atoms in another unit cell to form Cooper pairs. In this way the Cooper pairs of s-valence (and nonvalence) electrons of B, the Cooper pairs of s-valence electrons of Mg can be formed (In other words, through this channel opening the s-valence (or nonvalence) electrons of B and s-valence electrons of Mg can transit from the valence (or nonvalence) band to the conduction band from which Cooper pairs of these electrons can be formed by the attractive electron-electron interaction from the seagull vertex term) . Thus this channel opening gives the 3D region ( pi-band) of conventional superconductivity. From this 3D conventional superconductivity we have the existence of quasi-2D bifurcation region ( sigma-band) of high-Tc superconductivity given by the B plane. We notice that x o = 0 approximately. This agrees with the experiment that the state of superconductivity of Mg(l-x)Be(x)B2 is in the doping range 0 <= x < 1 and that MgB2 is in the state of superconductivity (c.f. J. Nagamatsu, et al., Nature, 410, 63 (2001) ; I. Felner, arXiv: cond-mat/0102508) . When (2) approximately holds, the s-valence electrons of B are in the state of first ionization energy of B. Then the s-nonvalence electrons of B are in the state of second ionization energy of B. Also the s-valence electrons of Mg are in the state of first ionization energy of Mg. The s-valence electrons of B and Mg in the state of first ionization energy are in the opened channel of 3D superconductivity while the s-nonvalence electrons of B in the state of second ionization energy of B are for the quasi-2D high-Tc superconductivity of the B plane. Thus the maximum value of the energy parameter h(Bl) and h(B2) of the s-valence and nonvalence electrons of B are proportional to the first and second ionization energies of B respectively, and the maximum value of the energy parameter h(Mg) of the s-valence electrons of Mg is proportional to the first ionization energy of Mg. Then, from the state of superconductivity h ² = 3 k ² , we have the following formula of Tc of MgB2:

k _B Tc = k = -ί- Delta(MgB2) (3)

3

where Delta(MgB2)=h=6h(Bl)+6h(B2) +h(Mg) is the energy gap of superconductivity of MgB2, where 6 = - is from the 12 valence or nonvalence s-electrons of the B atoms in a hexagonal unit cell of MgB2. From the table of ionization energies, the first and second ionization energies of B are approximately equal to 800.6 kJ/mol and 2427.1 kJ/mol. Let h(Bl)= xi 800.6 kJ/mol and h(B2)= xi 2427.1 kJ/mol, h(Mg)=xi 737.7 kJ/mol where xi = 2.83133971 10 ^~5 is a proportional constant determined from the computation of Tc of Nb. Then from (3) we have:

Tc = 39.53 K ( Computed value of Tc of MgB2) (4)

This agrees with the experimental value of Tc= 39 K of MgB2. We notice that the energy gap Delta(MgB2)=h=6h(Bl)+6h(B2) +h(Mg) contains the sum of two energy gaps Delta(l)=6h(Bl) and Delta(2)=6h(B2) . We have that Delta(l) correspond to the conventional 3D superconductivity and Delta(2) correspond to the quasi-2D high-Tc superconductivity. Thus we have the two-gap superconductivity. This agrees with the phenomenon of two-gap superconductivity of MgB2 (c.f. A.Y.Liu, et al., Phys.Rev.Lett., 87, 0870051 (2001) ; X.K.Chan, et al., Phys.Rev.Lett., 87, 1570021 (2001)) . In the following examples the principle of the doping mechanism of superconductivity applied to this MgB2 is applied to other intermetallics.

EXAMPLE 1. AlB2-type intermetallic Sr(l-x)Ca(x)Ga2

[0018] Let us consider an intermetallic of the form Sr(l-x)Ca(x)Ga2 ( 0 < x < 1 ) . We have that CaGa2 and SrGa2 can be formed in the AlB2-type phase as shown in L.H. Bennett and R.E. Watson, in Theory of Alloy Phase Formation, ed. L.H. Bennett, p.425 (The Metallurgical Society AIME 1980) . Thus the intermetallic Sr(l-x)Ca(x)Ga2 can also be formed in the AlB2-type phase, with the unit cell as shown in FIG.2 where the white balls representing Sr(Ca) (replacing Mg) and the black balls representing Ga (replacing B) .

[0019] For the doping mechanism of superconductivity let us consider the following function:

f(x)=6491x + 5500(l-x) (5) where 6491 kJ/mol and 5500 kJ/mole are approximately the fourth ionization energies of Ca and Sr respectively. This function gives the relation that the increasing of x gives the increasing of h. Then we set the following relation of the degenerate state of channel opening:

6491x ₀ + 5500(1- x ₀ ) = 6200 (6) for some x ₀ ( 0 < x o < l ) where 6200 kJ/mol is approximately the fourth ionization energy of Ga. A main point is that 6200 is close to 6491. When this relation holds (or approximately holds), the channel connecting the two states of fourth ionization energy of Ca and Ga can be opened. This channel opening gives a freedom of electric current with a direction orthogonal to the Ga plane. From this freedom of electric current, the Cooper pairs of the 3s,3p-level electrons of Ga and Ca can be formed (In other words, through this channel opening the 3s,3p-level electrons of Ga and Ca can transit from the valence band to the conduction band from which Cooper pairs of these electrons can be formed by the attractive electron-electron interaction from the seagull vertex term) . Thus this channel opening gives the 3D region of conventional superconductivity. From this 3D conventional superconductivity we have the existence of quasi-2D bifurcation region of high-Tc superconductivity given by the Ga plane.

[0020] We notice that x ₀ = 0.706 approximately. Then Sr(l-x)Ca(x)Ga2 comes into the range of high-Tc superconductivity when x o <= x <= xi for some xi such that 0 < x ₀ < xi <= 1 .

[0021] When (6) holds (or approximately holds) giving the channel opening, the 3s,3p-level electrons of Ga in the Ga plane are in the basic state of fourth ionization energy and the 3s,3p-level electrons of Ca are in the basic state of fourth ionization energy, and that other states are to be reached from these two states. Further the 3s,3p-level electrons of Ga and Ca are unified to occupy a sequence of states such that the 3s,3p-level electrons of Ga in the Ga plane occupy the higher states while the 3s,3p-level electrons of Ca occupy the lower states.

[0022] Thus, as MgB2, the 3s,3p-level electrons of Ga are in the state of fifth ionization energy of Ga; and the 3s,3p-level electrons of Ca are in the basic state of fourth ionization energy. The 3s,3p-level electrons of Ca in the basic state of fourth ionization energy are in the opened channel of 3D superconductivity, while the 3s,3p-level electrons of Ga in the state of fifth ionization energy are for the quasi- 2D high-Tc superconductivity of the Ga plane.

[0023] Then when (6) holds (or approximately holds) giving the channel opening, the Cooper pairs of the s,p- valence electrons of Ga and Ca can also be formed. These s- valence electrons of Ga and Ca are in the basic state of first ionization energy.

[0024] Thus, the maximum value of the energy parameter h(Ga5) of the 3s,3p-level electrons of Ga is proportional to the fifth ionization energy of Ga; the maximum value of energy parameter h(Ca4) of the 3s,3p-level electrons of Ca is proportional to the fourth ionization energy of Sr; and the energy parameters h(Ga) and h(Ca) of the s-valence electrons of Ga and Ca are proportional to the first ionization energies of Ga and Ca respectively. [0025] Then, from the state of superconductivity h ² = 3 k ² , we have the following formula of Tc of Sr(l-x)Ca(x)Ga2:

k — -Delta(CaSrGa) (7) where Delta(CaSrGa)=h=24 h(Ga5)+4 h(Ca4)+6 h(Ga)+h(Ca) is the energy gap of superconductivity of Sr(l-x)Ca(x)Ga2 where the coefficients 4, 6, 24=4 times 6 are from the 4 = | for the eight 3s,3p-level electrons of the Ga or Ca, and the 6 for the six Ga atoms of the hexagon of Ga. (For simplicity the effect of s- valence electrons of Sr and the effect of 3s, 3p- level electrons of Sr are omitted) .

[0026] Then we have h(Ga)= xi 578.8 kJ/mol, h(Ga5)= xi 8700 kJ/mol, h(Ca)=xi 587.8 kJ/mol and h(Ca4)= xi 6491 kJ/mol; and 8700 kJ/mol is approximately the fifth ionization energy of Ga, and 578.8 kJ/mol, 587.8 kJ/mol are approximately the first ionization energies of Ga and Ca respectively. Then from (7) we can compute the highest critical temperature Tc of Sr(l-x)Ca(x)Ga2 which is upped to:

Tc = 463.9 K (Computed Tc of Sr(l-x)Ca(x)Ga2) (8)

[0027] We may use other alkaline earth elements such as Ba (with fourth ionization energy 4700 kJ/mol) to replace the alkaline earth element Sr ( BaGa2 can also be formed in the AlB2-type phase) . The highest critical temperature Tc can also be upped to 463.9 K.

EXAMPLE 2. CaCu5-type intermetallic La(l-x)Ca(x)Cu5

[0028] Let us consider an intermetallic La(l-x)Ca(x)Cu5 (0 < x < 1) . We have that the intermetallics LaCu5 and CaCu5 are of the CaCu5-type as shown in D.J. Chakrabari and D.E. Laughlin, Bull. Alloy Phase Diagram, 2, 319 (1981) and P.R. Subramanian and D.E. Laughlin, Bull. Alloy Phase Diagram, 9, 316 (1988) . Thus the crystal structure of La(l- x)Ca(x)Cu5 can be formed in the CaCu5-type with La(l-x)Ca(x) corresponding to Ca.

[0029] This CaCu5-type is similar to the AlB2-type with the hexagon of six B atoms replaced by an enlarged hexagon of twelve Cu atoms and a hexagon of six Cu atoms is intercalated in one of the two Ca planes which replaces the corresponding one of the two Al planes of A1B2. There are eighteen Cu atoms near the faces of a unit cell of the hexagonal structure of CaCu5 (where twelve Cu atoms are from the two hexagons of six Cu atoms intercalated in the two Ca planes and six Cu atoms are from the enlarged hexagon of Cu) . FIG.3 shows the crystal structure of two of the three layers of a unit cell of La(l-x)Ca(x)Cu5 of CaCu5-type. Thus in a unit cell of CaCu5 nine Cu atoms of the twelve Cu atoms of the enlarged hexagon of Cu and six Cu atoms of the two hexagons of Cu and one face-centered Ca atom are as a cluster of atoms. For the doping mechanism of superconductivity let us consider the following function:

f(x)=1850(l-x)+4912 x (9) where 1850 kJ/mol and 4912 kJ/mole are approximately the third ionization energies of La and Ca respectively. This function gives the relation that the increasing of x gives the increasing of h. Then we set the following relation of the degenerate state of channel opening:

f(x ₀ ) = 1850 (1- x ₀ ) + 4912 x 1958 (10) for some x o ( 0 < x o < 1) where 1958 kJ/mol is approximately the second ionization energy of Cu (A main point is that 1850 is close to 1958 ) . When this relation holds (or approximately holds), the channel connecting the state of third ionization energy of La and the state of second ionization energy of Cu can be opened. This channel opening gives a freedom of electric current with a direction orthogonal to the Cu plane. By this freedom of electric current, the valence electrons of a cluster of atoms in a unit cell can be coupled to the valence electrons of other clusters to form Cooper pairs. In this way the Cooper pairs of d- valence electrons of Cu and the Cooper pair of the s-valence electrons of La can be formed (In other words, through this channel opening the d- valence electrons of Cu can transit from the valence band to the conduction band from which Cooper pairs of these d-valence electrons can be formed by the attractive electron-electron interaction from the seagull vertex term) . Thus this channel opening together with the Cu plane gives the region of 3D conventional superconductivity. From this region of 3D superconductivity we have the existence of quasi-2D bifurcation region of the high-Tc superconductivity given by the Cu plane. We notice that x o = 0.035 approximatly. Then La(l-x)Ca(x)Cu5 comes into the range of superconductivity when x o < x < xi for some xi such that x o < xi < 1. When (10) holds (or approximately holds) giving channel opening, the d-valence electrons of Cu in the Cu plane are in the basic state of second ionization energy and the d-valence electron of La is in the basic state of third ionization energy, and that other states are to be reached from these two states. Further the d-valence electrons of Cu in the Cu plane and the d-valence electron of La are unified to occupy a sequence of states such that the d-valence electron of Cu in the Cu plane occupy the higher states while the d-valence electron of La occupy the lower state. Thus, as MgB2, the d-valence electrons of Cu are in the state of third ionization energy, the d-valence electron of La is in the state of third ionization energy; and the s-valence electrons of La are in the basic state of first ionization energy.

[0030] Thus, the maximum value of the energy parameter h(Cu) of the d-valence electrons of Cu is proportional to the third ionization energy of Cu, and the energy parameter h(La) of the s-valence electrons of La is proportional to the first ionization energy of La. Then, from the state of superconductivity h ² = 3 k ² , if we omit the ef feet of the hexagon of six Cu atoms intercalated in the La plane, we have the following formula of Tc of La(l-x)Ca(x)Cu5:

k _B Tc = k = -i- Delta(LaCaCu) (11)

3 2

where Delta(LaCaCu)=h=45 h(Cu) + h(La) is the energy gap of the superconductivity of La(l-x)Ca(x)Cu5 where the coefficient 45=5 times 9 is from the 5 = -ψ for the ten d-valence electrons of the Cu, and the 9 for the nine Cu atoms in the enlarged hexagon of Cu. (For simplicity the effect of s, d-valence electrons of Ca is omitted) . We have h(Cu)= xi 3555 kJ/mol, and h(La)= xi 538.1 kJ/mol where 3555 kJ/mol is approximately the third ionization energy of Cu and 538.1 kJ/mol is approximately the first ionization energy of La. Then from (11) we can compute the highest critical temperature Tc of La(l-x)Ca(x)Cu5 which is upped to:

Tc = 315.6 K ( Computed Tc of La(l-x)Ca(x)Cu5) (12) If we include the effect of the hexagon of six Cu atoms intercalated in the La plane and suppose that this effect is the same as the enlarged hexagon of twelve Cu atoms. Then the term 45h(Cu) in Delta(LaCaCu) becomes 75h(Cu) . Then the highest critical temperature Tc of La(l-x)Ca(x)Cu5 is upped to:

Tc =525.3 K ( Computed Tc of La(l-x)Ca(x)Cu5) (13)

[0031] We may use other alkaline earth elements such as Sr (with the third ionization energy 4138 kJ/mol) to replace Ca to form the intermetaUic La(l-x)Sr(x)Cu5 ( SrCu5 is also of the CaCu5-type) . The doping mechanism is similar to La(l-x)Ca(x)Cu5 with x o= 0.047 approximately. The highest critical temperature Tc can be upped to 525.3 K. We have that La(l-x)Sr(x)Cu5 is easier to be synthesized than La(l-x)Ca(x)Cu5 since 4138 < 4912.4 . Further, we may use a mixture of Ca and Sr, denoted by A, to replace Ca to form the intermetaUic La(l-x)A(x)Cu5. Also we may use a mixture of Cu and Zn to replace Cu to form the intermetaUic La(l-x)A(x)Cu5(l-y)Zn(5y) . Also we may use rare earth elements R and mixture thereof to replace Ca to form the intermetaUic La(l-x)R(x)Cu5. Also we may replace La with rare earth elements R (including the element Y and mixture thereof) to form intermetaUic R(l-x)A(x)Cu5 and to get superconductors by doping this intermetallics to be in the degenerate state of channel opening.

EXAMPLE 3. CaCu5-type intermetaUic LaNi(5-5x)Cu(5x)

[0032] Let us consider an intermetaUic LaNi(5-5x)Cu(5x) ( 0 <= x <= 1) . We have that the intermetallics LaCu5 and LaNi5 are of the CaCu5-type. Thus LaNi(5-5x)Cu(5x) can be formed in the CaCu5-type with Ca corresponding to La and Cu corresponding to Ni(l-x)Cu(x) . For the doping mechanism of superconductivity let us consider the following function:

g(x)=1753(l-x)+3555x (14) where 3555 kJ/mole is approximately the third ionization energy of Cu. This function also gives the relation that the increasing of x gives the increasing of h. Then we set the following relation of the degenerate state of channel opening:

g(x ₀ ) = 1753(1- x ₀ ) + 3555x o = 1850 (15) for some x o ( 0 < x o < 1) where 1850 kJ/mol is again approximately the third ionization energy of La (The ionization energy of Cu can be the second or the third ionization energies) . When this relation holds (or approximately holds) we also have that the channel connecting the state of third ionization energy of La and the state of second ionization energy of Ni(Cu) can be opened. Then we have x o= 0.0538 approximately. Then LaNi(5-5x)Cu(5x) comes into the range of superconductivity when x o <= x <= xi for some xi such that x o < xi < 1. Since x o = 0 approximately, we have that LaNi5 can be formed in the degenerate state of channel opening. Thus LaNi5 can be formed as a superconductor. When (15) holds (or approximately holds) giving channel opening, the d-valence electrons of Ni in the Ni(Cu) plane are in the basic state of second ionization energy and the d-valence electrons of La are in the basic state of third ionization energy, and that other states are to be reached from these two states. Further the d- valence electrons of Ni in the Ni(Cu) plane and the d- valence electron of La are unified to occupy a sequence of states such that the d- valence electrons of Ni in the Ni(Cu) plane occupy the higher states while the d-valence electron of La occupies the lower state. Thus, as MgB2, the d-valence electrons of Ni are in the state of third ionization energy, the d-valence electron of La is in the state of third ionization energy; and the s- valence electrons of Ni are in the state of first ionization energy, the s- valence electrons of La are in the state of first ionization energy. Thus, the maximum value of the energy parameter h(Ni) of the d-valence electrons of Ni is proportional to the third ionization energy 3395 kJ/mol of Ni, the energy parameter h(Nil) of the s- valence electrons of Ni is proportional to the first ionization energy 737.1 kJ/mol of Ni, and the energy parameter h(La) of the s- valence electrons of La is proportional to the first ionization energy of La. Then, as La(l-x)Ca(x)Cu5, we have the following formula of Tc of LaNi(5-5x)Cu(5x) :

k _B Tc = k = -! _r Delta(LaNiCu) (16)

3

where Delta(LaNiCu)=h=60h(Ni)+15 h(Nil)+ h(La) is the energy gap of LaNi(5(l-x))Cu(5x) and 60=4 times 15 where 4 = | is from the eight d-valence electrons of Ni, and the 15 for the fifteen Ni(Cu) atoms of the cluster of Ni(Cu) atoms in a unit cell (For simplicity the effect of d-valence electrons of Cu is omitted) . Then from (16) we can compute the critical temperature Tc of LaNi(5(l-x))Cu(5x) which is upped to:

Tc = 423.321 K ( Computed Tc of LaNi(5(l-x))Cu(5x) ) (17)

We may use other transition elements such as Co to replace Ni or Cu of LaNi(5(l-x))Cu(5x) .

[0033] Since x ₀ = 0 approximately, we have that LaNi5 can be formed in the degenerate state of channel opening and thus LaNi5 can be formed as a superconductor with Tc upped to 423.321 K. This intermetallic LaNi5 had been used for hydrogen storage since LaNi5 is easy to be activated and can store a large amount of hydrogen under ambient pressure and in the room temperature. Since the activation and the hydrogen storage of a material is from the activity of electrons of this material, this property of LaNi5 shows that the d-valence electrons of Ni and La in LaNi5 are in the degenerate state of channel opening.

[0034] We may replace La with a mixture of rare earth elements such as Mm where Mm denotes a Mischmetal which is a mixture of rare earth elements with a large fractional part of La and the fractional part of Ce is more than | . Then as LaNi5 we have that MmNi5 is a superconductor with Tc > 300 K.

EXAMPLE 4. CaCu5-type intermetallic Sr(l-x)Ca(x)Cu5

[0035] Let us consider an intermetallic Sr(l-x)Ca(x)Cu5 ( 0 < x < 1) . This intermetallic is similar to the above intermetallic Sr(l-x)Ca(x)Ga2 with Ga2 replaced by Cu5, and is a special case of the above CaCu5-type intermetallic R(l-x) A(x)Cu5 with x=l ( A denotes a mixture of Ca and Sr) . We have that CaCu5 and SrCu5 can be formed in the CaCu5-type phase. Thus Sr(l-x)Ca(x)Cu5 can also be formed in the CaCu5-type phase. For the doping mechanism of superconductivity let us consider the following function:

f(x)=5500(l-x)+6491x (18) where 6491 kJ/mol and 5500 kJ/mole are approximately the fourth ionization energies of Ca and Sr respectively. This function gives the relation that the increasing of x gives the increasing of h. Then we set the following relation of the degenerate state of channel opening:

5500 (1- x ₀ ) + 6491x o 5536 (19) for some x o ( 0 < x o < 1) where 5536 kJ/mol is approximately the fourth ionization energy of Cu. When this relation holds (or approximately holds) , the channel connecting the two states of fourth ionization energy of Sr and Cu can be opened. This channel opening gives a freedom of electric current with a direction orthogonal to the Cu plane. From this freedom of electric current, the Cooper pairs of the 3s,3p-level electrons of Cu and the 4s,4p-level electrons of Sr can be formed. Thus this channel opening gives the 3D region of conventional superconductivity. From this 3D conventional superconductivity we have the existence of quasi-2D bifurcation region of high-Tc superconductivity given by the Cu plane. We notice that x o= 0.0363 approximately. Thus Sr(l-x)Ca(x)Cu5 comes into the range of superconductivity when x ₀ < x < Xi for some x i such that x ₀ < Xi < 1. When (19) holds (or approximately holds) giving channel opening, the 3s,3p-level electrons of Cu in the Cu plane are in the basic state of fourth ionization energy and the 4s,4p-level electrons of Sr are in the basic state of fourth ionization energy, and that other states are to be reached from these two states. Further the 3s,3p-level electrons of Cu and the 4s,4p-level electrons of Sr are unified to occupy a sequence of states such that the 3s, 3p- level electrons of Cu in the Cu plane occupy the higher states while the 4s,4p-level electrons of Sr occupy the lower states. Thus, the 3s, 3p- level electrons of Cu are in the state of f ifth ionization energy of Cu; and the 4s,4p-level electrons of Sr are in the basic state of fourth ionization energy. The 4s,4p-level electrons of Sr in the basic state of fourth ionization energy are in the opened channel of 3D superconductivity, while the 3s, 3p- level electrons of Cu in the state of f ifth ionization energy are for the quasi-2D high-Tc superconductivity of the Cu plane. Then when (19) holds giving channel opening the Cooper pairs of the s,p-valence electrons of Sr can also be formed. These s- valence electrons of Sr are in the basic state of first ionization energy (and can be in the states of third and second ionization energies of valence electrons respectively) .

[0036] Thus, the energy parameter h(Cu5) of the 3s,3p-level electrons of Cu is proportional to the f ifth ionization energy of Cu; the energy parameter h(Sr4) of the 4s, 4p- level electrons of Sr is proportional to the fourth ionization energy of Sr; and the energy parameter h(Sr) of the s- valence electrons of Sr is proportional to the f irst ionization energies Sr respectively (when the s- valence electrons of Sr is in the basic state of first ionization energy) . Then, as Sr(l-x)Ca(x)Ga2, we have the following formula of Tc of Sr(l-x)Ca(x)Cu5: k — Delta(CaSrCu) (20) where Delta(CaSrCu)=h=60 h(Cu5)+4 h(Sr4)+h(Sr) is the energy gap of superconductivity of Sr(l-x)Ca(x)Cu5 where the coef f icients 4, 60=4 times 15 are from the 4 = | for the eight 3s,3p-level electrons of the Cu or the eight 4s,4p-level electrons of the Sr, and the 15 for the fifteen Cu atoms of the CaCu5-type structure. We have h(Cu5) = xi 7700 kJ/mol, h(Sr) = xi 549.5 kJ/mol and h(Sr4) = xi 5500 kJ/mol where 7700 kJ/mol is approximately the fifth ionization energy of Cu, and 549.5 kJ/mol is approximately the first ionization energy of Sr. Then from (20) we can compute the highest critical temperature Tc of Sr(l-x)Ca(x)Cu5 which is upped to:

Tc = 952.7 K (Computed Tc of Sr(l-x)Ca(x)Cu5) (21)

[0037] As the above intermetallic R(l-x)A(x)Cu5, We may replace Cu with TM=Ni, Co, Zn and the mixture thereof to form Sr(l-x)Ca(x)TM5) .

[0038] Then we notice that the intermetallic R(l-x) A(x)Cu5 where R denotes a rare earth element including the element Y and the mixture thereof may have two cases of high-Tc superconductivity: a case is the case of the Cu d-electron (from the effect of both A and R) and the other case is the case of the Cu s,p-electron (from the effect of A only or from the effect of both A and R) . When both cases of superconductivity appear at some doping, the effects of superconductivity of these two cases can be combined. From this combination of effects of superconductivity we can have a larger critical current consisting of d-electrons and s,p-electrons and a higher critical temperature Tc. As an example let us consider the intermetallic La(l-x)Sr(x(l-y))Ca(xy)Cu5 ( 0 < x, y < 1) . Let the doping parameters x,y

(l-y)5500 +y6491 5536

When this relation of the degenerate state of channel opening holds (or approximately holds), the s,p-channel connecting the two states of fourth ionization energy of the 3s,3p-electrons of Cu and the 4s,4p-electrons of Sr can be opened, and also the d-channel connecting the two states of ionization energy of the 3d-electrons of Cu and the 5d-electron of La can be opened. This two-channel-opening gives a freedom of electric current with a direction orthogonal to the Cu plane. From this freedom of electric current, the Cooper pairs of the 3s, 3p, 3d-electrons of Cu and the 4s,4p-electrons of Sr can be formed. Then, this two-channel- opening can give two types of high-Tc superconductivity: the s,p-electron superconductivity and the d-electron superconductivity. This combination of s,p-electron superconductivity and d-electron superconductivity can give larger critical current and higher critical temperature.

A GENERAL FORM OF: CaCu5-type Intermetallics

[0039] We notice that the hexagonal unit cells of the above examples of CaCu5-type intermetallics are with a large cluster of superconducting electrons, thus these CaCu5-type intermetallics are with high critical current density of superconducting current. Also we notice that the doping of the above CaCu5-type (or AlB2-type) intermetallics gives the degenerate state of channel opening for superconductivity. Thus the doping of the above CaCu5-type (or AlB2-type) intermetallics has the effect of introducing pinning centers of superconducting current to these CaCu5-type (or AlB2-type) intermetallics.

[0040] We may apply more dopings to form more CaCu5-type intermetallics. For superconductivity these dopings must give the degenerate state of channel opening. By more dopings we have the following form of intermetallic which includes the above examples of CaCu5-type intermetallics:

R(l-x)A(x) [TM(5(l-y))M(5y)] (l+z) (23) where 0 <= y <= y ₀ 0 <= z <= for some small parameters y ₀ , z o (The doping parameters y ₀ , z o are small for keeping the CaCu5-type phase) ; TM=Cu, Ni, Co, Zn, and M is a mixture of elements Ti, Cr, Mn, Fe, Co, Zr, Nb, Mo, Hf, Ta, W, Ga, In, Al, Si, Ge, Sn. These elements are with states of second or third ionization energies quite close to that of R or TM. Thus for some chosen M, y and z, this intermetallic can give the degenerate state of channel opening.

[0041] From the properties of the CaCu5-type intermetallic (23) we have a process of manufacturing CaCu5-type superconductors, as follows. The powders of constituent of a CaCu5-type intermetallic (23) are first mixed in accordance with the composition. Then the mixture of powders are melted in an induction furnace under argon atmosphere at a temperature between 800 ° C. and 1600 ° C. After the materials are melted, the melt is maintained at the same temperature for 20 minutes to 1 hour to achieve better homogeneity. The melt is then poured into a mold and under pressure between the ambient pressure and 6 GPa cooled down for solidification and for the formation of said CaCu5 phase. After cooling, under pressure between ambient pressure and 6 GPa the resulting CaCu5-type ingot is annealed in a furnace at a temperature between 600 ° C. and 1200 ° C. in an argon atmosphere for 0.5 to 24 hours to achieve the state that the 3d-level (or 3s,3p-level) electrons of TM=Cu, Ni, Co, Zn and the mixture thereof are in the degenerate state of channel opening. This gives the bulk form of the intermetallic (23) with superconductivity.

A GENERAL FORM OF: Intermetallics with Hexagonal Crystal Structure

[0042] We can express the above examples of CaCu5-type and AlB2-type superconductors into the following general form of intermetallics:

R (l-x ⁽¹⁾ )E ((l-x KDim (1-x ⁽³⁾ ))G( mx ⁽³⁾ )) (l- y)M(my)] (l+z) (24) wherein R denotes rare earth elements (including Y) and mixture thereof, E and A denote alkaline earth elements and mixture thereof and E not equal A when x ¹ - ¹ - ¹ = 1, such that said R, E, A elements are in the upper and lower layers of the hexagonal crystal structure of three layers of (24) and D=Cu, Ga, Al, Si, Ni, Co, Zn and G=Cu, Ga, Ge, Si, Ni, Co, Zn are elements in the middle layer forming a conducting layer and may also be in the upper and lower layers, and m=2 or 5, and x W, χ (2) , χ (3) are doping parameters such that 0 <= x W <= 1, 0 < χ ( ² ) < 1, 0 <= χ (¾ <= 1; and M=Ti, Cr, Mn, Fe, Co, Zr, Nb, Mo, Hf, Ta, W, Al, Ga, In, Si, Ge, Sn and mixture thereof, and y, z are small positive parameters.

[0043] When m=2 this intermetallics (24) is formed with the AlB2-type crystal structure and when m=5 this intermetallics (24) is formed with the CaC5-type crystal structure. For this intermatallic (24) to be in the state of superconductivity, this intermetallic (24) is doped to be in the degenerate state of channel opening as specified in the above examples.