Kopp et al., [28] noted that any trace of AFM fluctuation is absent in over doped regime of cuprates. Supercon- ductivity arises in Mott insulators after doping level of nearly 5%; attains optimal value at 16%; overdoping starts at 19% and the superconducting dome terminates in the vicinity of 25%. In the overdoped regime, spin susceptibility shows a ferromagnetic upturn. He also suggested that at the end of superconductivity dome, there should be genuine ferromagnetism at zero temperature. Sonier et al. [29] observed gradual disappearance of an- tiferromagnetism on doping of cuprates and onset of static magnetic order in the highly doped regimes. But this magnetism is not of long range order but magnetic moments appear in dilute form. The main point in [28] [29] is that charge doping or hole-doping induces FM order in cuprates and competing ferromagnetic fluctuations are simultaneously present with superconductivity. It has also been concluded that on increasing doping level upto 25%, ferromagnetic fluctuations increase at the expense of antiferromagnetic fluctuations.

We have proposed models of preformed hole-pairs in cuprates [3] - [5] on the basis of FM spin fluctuation in (CuO)₄ plaquette of CuO₂ plane, but there is plethora of evidence that AFM fluctuations are intrinsic properties of cuprates. We will now show that hole-pairs can be formed on the same lines as done earlier without consid- eration of AFM fluctuations. For this, let us consider only A, B, C cells of figure 1, shown separately in figure 2.

But here a question arises, why in neutron diffraction experiments on cuprate superconductors, only AFM fluctuations are observed and not FM fluctuations. To answer this question, let us calculate the percentages of FM and AFM alignments of spins during one time-period of journey of Holes 1 and 2 in figure 1. The cal- culation of FM and AFM order can be done under 2 assumptions: 1) in whichever 2 cells out of the 9 cells the 2 holes are present, they will be in FM alignment and the rest 7 in AFM alignment. In this way, during one time-period of journey of two holes in figure 1, will be in FM order and in AFM order.

Under the assumption (2) when Holes 1 and 2 are in cells A and E respectively in the beginning, there will be FM order in cells A and E and in the rest 7, AFM order. When Hole 1 enters cell B and Hole 2 enters cell D, there will be subdued FM order in cells A and B due to Hole 1 and in cells E and D due to Hole 2, because there is common boundary between A and B for Hole 1 and common boundary between E and D for Hole

2. It can be said that there is half FM order both in A and B due to Hole 1 and half FM order both in D and E due to Hole 2. Half FM order is justified, because when both the holes enter cell C, FM order is nearly lost. It can be said that full ferromagnetism is due to the hole1 while in A and no ferromagnetism while in C. Thus when Hole 1 has reached cell B from A, it can be appropriately said that in both cells there is ferromagnetism of half strength only. From the cell C, due to curvature of the paths of holes and Coulomb repulsion, Hole 1 is re- flected towards cells H-I and Hole 2 towards G-F. When Holes 1 and 2 reach cell H and G respectively, half FM order is attained, but FM order is built downward of the CuO₂ plane. When Holes 1 and 2 reach I and F cells re- spectively, there will be full FM order downward of the CuO₂ plane. When the holes turn back from the cells I and F at the extreme ends, they enter cells H and G respectively, half FM order is obtained in cells I and H due to Hole 1 and in cells F and G due to Hole 2. Again both the holes reach cell C, where there will be negligible or zero magnetic order. From there, Hole 1 will be reflected towards cells D-E and Hole 2 towards cells B-A with the same type of magnetic order as in going from cell C towards cells H-I for the Hole 1 and towards G-F for Hole 2, but FM order now building upward of the CuO₂ plane. When Hole 1 has reached cell E and Hole 2 cell A, full FM order will be attained and by this time their journey in one time-period T has been completed. The- reafter they will be following the same path repeatedly.

Calculations by the first assumption gives percent of cells in FM spin alignment and percent in AFM alignment. Calculation by the second method gives 16% FM order and AFM order. Calculation of FM and AFM orders have been shown in table 2. Ratio of FM order to AFM order. Calculation from the second assumption is more realistic. But the above calculations have been done in the most favourable case where at all the possible places of hole-pair formation, holes are present. In actual cases (from underdoped to optimum doped), there will be many patches where hole-pairs would not have been formed be- cause no holes are available. Because of the small percentage of FM alignment in comparison to AFM align- ment, the former may be submerged under AFM alignment in actual experiments. AFM coupling of spins of holes of Cu²⁺ spins is unable to explain superconductivity. But FM coupling of Cu²⁺ spins can explain super- conductivity [4] but not supported by neutron scattering experiments. The reason may be that under overwhelm- ing AFM order, minority transitory FM order is submerged. The present mechanism of hole-pair formation is supported by theoretical considerations [4] and cannot be rejected by experiments because of its indetectibility of FM order due to its lean and fleeting presence. Time-period of a hole in Y-123 is. And in this time a hole has to cross 10 cells (not only 9 cells but 10 cells because C cell will come twice in its path). Time spent in C cell will be greater than in any other cell, because in this cell both of the holes face Coulomb repulsion and their velocities are reduced and practically there is no magnetic order in this cell. So a hole has to pass through 8 cells only with some kind of FM order and it will take less than per cell. There is another reason for FM order not to be observed in experiments is that half of time, projections of holes will be above the plane and half of time below the plane. They may cancel each other because changeover is very fast. It may be possible that the transitory FM order in such a shot spell is not detected in neutron scattering experiments. Kopp et al. [28] and Sonier et al. [29] have maintained that in superconductivity dome in the phase diagram, FM order coexists with AFM order.

Formation of hole-pairs in cuprates with large coherence lengths in a-b plane: We have till now dis- cussed formation of preformed hole-pairs taking Y-123 as a representative example which has coherence length of the order of 15 - 20 Å. But there are cuprate superconductors whose are much larger than that of Y-123. There are also cuprates whose are nearly equal to that of Y-123 or a little smaller. They can be un- derstood on the lines of arguments given for Y-123. The problem is how to explain hole-formation when is quite large. A broad view of the properties of cuprate superconductors is given in the table 3with the parameters with which we may be concerned in this paper. From table 3, one thing becomes clear that for a single CuO₂ layered superconductor, has the largest value, followed by two layered and the smallest ones are for the three layered cuprates. Coherence lengths along c-axis are quite small in all cases. For small values, Kumar et al. [30] have given a reasonable explanation for highly enhanced resistivity in c-axis transport in normal state. They interpreted the suppression of single particle transport along the c-axis in the normal state due to the blocking of inter-block transport by the intra-block coupling to many-body environments (i.e., entanglement with other electronic degrees of freedom). This mechanism is called Quantum Zeno Effect (QZE). From this it can be said that coherence lengths depend on the velocity of transport of charge carriers. Regarding the values of it can be said that when there are more than one CuO₂ layer, there is some kind of interaction between lay- ers that reduces the velocity of charge carriers which lowers the value of in two CuO₂ layered and the lowest value in three layered cuprates. Example of interaction between layers of CuO₂ in cuprate superconduc-

tors is the occurrence of odd and even AFM excitations in two CuO₂ layered cuprates. It explains why is highest in single layered cuprates. As in multi-layered cuprates, in single layered cuprates also velocity of holes

depends on the buckling of Cu-O-Cu angle in the CuO₂ plane. Buckling may arise due to interaction of

and orbitals of the oxygen ions of nearest layers vertically above or below. Interaction between

orbital of the Cu²⁺ ion and and orbitals of the planar oxygen when a hole reaches O²⁻ ion may be another cause of buckling.

It has been found that CuO₂ plane in TL-2201 is quite flat and smooth. The effect of smoothness of CuO₂ plane can also be guessed from comparison of the velocities of hole pairs in Y-123 and Tl-2201. In Y-123, each hole of a hole-pair covers angular distance corresponding to linear distance of Å (each side of the cell taken to be equal to Å). In the case of Tl-2201, the linear dis- tance Å. Velocity of hole-pair = distance travelled/time period. For Y-123, velocity.

For Tl-2201, velocity. Thus velocity of hole pairs in Tl- 2201 is times more than that in Y-123. It is known that velocity of hole-pairs in cuprates is proportional to (where superconducting density and effective mass of hole pair). All the data about Y- 123 and Tl-2201 used here have been taken for optimum doping, hence n_s will be nearly equal in both cases. So velocity of hole-pairs should be proportional to. Thus it can be concluded that the coherence length of cuprates depends upon the effective mass of hole-pair in superconductors. It may also be mentioned here that effective mass of hole-pairs is less than that of single hole.

Different effective masses of hole-pairs in different cuprates may be attributed to the nature of near neigh- bours of CuO₂ plane in the vertical direction. In Y-123, near neighbours are as follows: CuO-BaO-CuO₂-Y- CuO₂-BaO-CuO and for Tl-2201, they are as follows: TlO-TlO-BaO-CuO₂-BaO-TlO-TlO. Near neighbours in the two cases are quite different.

Near neighbours control the smoothness and flatness of the plane or buckling of Cu-O-Cu angle which ulti- mately controls the velocity of holes in the CuO₂ plane. One difference that is obvious in the immediate neigh- bourhood of CuO₂ plane in Tl-2201 and Y-123 is that in the former, Cu²⁺ ions of CuO₂ plane have bipyramidal coordinations with the O²⁻ ions of BaO plane, whereas in the latter, on one side of CuO₂ plane , there is coordi- nation between Cu²⁺ ions of CuO₂ plane with O²⁻ ions of BaO plane but on the other side, Cu²⁺ ions of CuO₂ plane are not coordinated to any oxygen ion , because in Y layer there is no oxygen ion. In addition to coordina- tion of ions, the atoms of surroundings may affect the properties of CuO₂ layers by differences in electronic structure, ionization energy and electronegativity etc.

The general condition for Heisenberg exchange interaction to take place is that the ratio must be greater than 3 but not much greater, where distance between atoms i and j atoms and radius of the 3d orbital. But there are instances where Heisenberg exchange interaction is effective at much larger separations. In a series of free radicals in aqueous solution, the exchange rate [33] is much greater than the reaction rate and the critical exchange distance is between one and three hard sphere encounter distance in agreement with several theoretical predictions. In [33] , exchange interaction has been found to be effective at distances Å. Veloc- ity of a hole in Tl-2201 is much higher than that in Y-123. Higher velocity means that the surface is flat and smooth which means that there is no or very small variation of electron density along its path or the buckling angle is zero or very small. When intervening space between two holes is smooth or without any variation of electron density, the wave functions of electrons can spread over large distances. Holes situated at much larger distances than can also be bound by Heisenberg exchange interaction. It can be concluded that hole-pair formation in Tl-2201 (with very large) and Y-123 (with small or moderate) can be explained on the same lines by transitory FM order induced by holes wandering in the CuO₂ plane.