Numerical Approach in Superconductivity: An advanced study

. The dependence of the critical temperature T c of high-temperature superconductors of various families on their composition and structure is proposed. A clear dependence of the critical temperature of high-temperature superconductors (hydrides, Hg- and Y-based cuprates) on the serial number of the constituent elements, their valence and crystal lattice structure has been revealed. For Hg-based cuprates, it is shown that it is possible to obtain even higher temperatures of superconducting transitions at normal pressure by implanting mercury atoms into the crystal lattice of cuprate.


Introduction
Recent results in the field of high-temperature superconductivity have aroused great interest of the scientific community to search for a theoretical explanation of the mechanism of the appearance of a superconducting state at temperatures close to room temperature [1], [2]. However, many theoretical questions remain unresolved, while the experiment approached room temperature at high pressures and halfway to room temperature at normal pressures.
Various numerical approaches have been used to describe superconductivity. For example, numerical approach in description of vortex nucleation in the Ginzburg-Landau model of superconductivity [3] and a numerical method of solving the mean-field self-consistent Bogoliubov de Gennes equations by expanding the Nambu Greens functions in terms of Chebyshev polynomials [4] which both require phenomenological assumptions or complex calculations on grid. All of them do not give a clear dependence of the observed parameters of a superconductor on its material composition and lattice structure. In the article [2], good agreement was obtained between the critical temperature of the superconducting transition and the parameters of high-temperature superconductors, but these parameters themselves require experimental measurements and have a complex dependence on the composition and structure of the sample, which is still questionable. This study uses the basic idea that there is a clear relationship between the content and structure of a superconductor and its critical superconducting transition temperature, expressed in integer and rational numbers, as is the case for the laws of quantum mechanics.
An attempt is made in this work to establish a relationship between the superconducting transition temperature T c of various families of high-temperature superconductors on the composition and structure of the superconductor material. For this, a numerical approach is used, similar to Balmer's work using number theory, which was only once applied Composition: to fundamental problems of physics over a hundred years ago when describing the emission spectra of hydrogen [5]. His result was confirmed by Rydberg [6] and explained by Bohr when he was creating quantum mechanics [7].

Method and Results
The essence of the applied method is as follows. Consider Hg-based cuprates Hg m Ba 2 Ca n−1 Cu n O 2n+m+2 or another notation Hg − m2(n − 1)n. Suppose that T c is proportional to the combination of integers, ordinal numbers of the elements of a given cuprate, and the number of elements in the chemical formula of the cuprate. Using the enumeration method (enumeration of various dependences of the critical temperature on the serial numbers of chemical elements of the substance), it turned out that the following combination is in good agreement with the experimental value of T c :  Table 1 shows the transition temperatures calculated using the formula (1) and taken from the literature.
Further, we assume that the crystal lattice remains the same (we do not change all the coefficients in the formula (1)), but we introduce mercury atoms into the crystal cell -we increase the coefficient at Hg by one, leaving m unchanged: In the table 1, the calculation by the formula (2) is shown in the 4th column for (hypothetical) samples with mercury atoms embedded in the crystal lattice. For material with m = 1, an increase in T c is visible. Coincidence of T c = 152 K for Hg-1223, calculated from (2), with the transition temperature of the Hg-1223 sample at a pressure of P 0 = 150 kbar [10] can be considered not accidental if we assume that the introduction of mercury atoms into the crystal cell of Hg-1223 creates mechanical stresses equivalent to those at a given external pressure P 0 .
Considering the position of the nodes of the crystal lattice of Hg-1223 unchanged upon the introduction of mercury atoms (from one to eight penetrating atoms) into it, we will make an approximate estimate of the magnitude of mechanical stresses in the crystal cell of Hg-1223. Crystal cell parameters of Hg-1223: a = 3.85Å, c = 15.8Å [11].
Total energy of elastic deformation of the crystal cell [12]: where G = 27.6 GP a -shear modulus is taken for another cuprate T l 2 Ba 2 CaCu 2 O 8+δ [13], since for cuprates based on Hg these modules have not yet been found in the literature; r 0 = 1Å -radius of the sphere in the cell where the mercury atom is introduced (preliminary estimate); r 1 = 1.55Å is the radius of the mercury atom. = 80 kbar, which coincides in order of magnitude with P 0 . Thus, it was shown that the incorporation of mercury atoms into the lattice cells of the sample Hg-1223 creates mechanical stresses in the crystal lattice, equivalent to those that arise at increased external pressures, into which a common sample of Hg-1223 is placed.
Further analysis of formula (1) revealed that the combination 300 · 2/C 0 coincides with the average temperature T c for α− and β−modifications of mercury: T Hg c = 300 · 2/C 0 = 4.074 K [14], [15]. Then formula (1) can be rewritten in the form: where the superscript at T c simply denotes the element, but not the power. Using the same approach as for cuprates, and taking into account the last nuance regarding the nature of the constant C 0 , a formula was obtained that establishes the relationship between the superconducting transition temperature and the material composition of high-temperature superconductors of the family of hydrides AH n or AD n , where A = S, La, Y, P r, P, Si, Ba, T h: where T A c is the critical temperature of the element A (table 2), and the superscript A is just a designation of the element, not an exponent; in brackets A -serial number (nuclear charge) of the element; n -an integer from the material designation -the number of hydrogen or deuterium atoms; h -integer or rational number depending on the space group of the crystal lattice of the sample (table 2); N is the number of elements in the substance -everywhere N = 2, because there are always two elements: A and hydrogen or deuterium. Table 2 shows the transition temperatures calculated using the formula (5) and taken from the literature. The value of h (in table 2) has a specific meaning for each space group and type of crystal lattice -atomic, ionic, metallic or molecular. There are not enough statistics yet to write an exact formula for h. In the formula (5), the effect of pressure is taken into account by choosing the value of the critical temperature of component A at different pressures at which the calculation and experiment coincide best of all (if such T A c are given in the literature). Otherwise, T A c is taken at atmospheric pressure. In the case of praseodymium, there are no data at ultrahigh pressures of the order of 100 GPa and more. Therefore, Pr can be expected to be a superconductor with a critical temperature of about 2 K at about 100 GPa or more.
As a next step, based on the previous results, a formula was obtained that establishes the relationship between the superconducting transition temperature and the material composition of high-temperature superconductors of the yttrium-based cuprate family Y m Ba k Cu n O p : where T Y c = 9.27 K is the critical temperature for yttrium at 31 GPa [31]; m, k, n and p are integers from the material designation; designation of an element -its serial number, or nuclear charge, i.e. Y = 39, Ba = 56, Cu = 29, O = 8; N is the number of elements, not counting Cu-O: i.e. N = 2; n a is the number of Cu − O planes in the crystal cell [32] (always equal to 2). Table 3 shows the transition temperatures calculated using the formula (6) and taken from the literature.
When outputting (6), other values of T Y c were also used: T Y c = 7.9 K at 26 GPa [31] and T Y c = 1.5 − 2.7 K at 12-16 GPa [33], but the formulas obtained with a different combination of numbers in front of the brackets of (6) gave much worse agreement with the experiment. This suggests that the mechanical stresses in the crystal cell of Y-based cuprates are close to 30 GPa, at least near the Y atom.

Conclusion
Using a numerical approach, formulas are obtained that relate the superconducting transition temperature to the composition and structure of the material for various families of high-temperature superconductors. A prediction has been made regarding the method of increasing T c for Hg-based cuprates by implanting Hg atoms into the crystal lattice of the cuprate, which will create mechanical stresses in the lattice equivalent to stresses at increased external pressure.
It is not surprising that T c would have a systematic dependence on integers, since the consensus view of superconductivity, regardless of physical instantiation, is that of Cooper pairs, formed via the exchange of a boson (phonon). The pairing will depend in a systematic way on the coupling to this boson, the characteristic energy of this boson, and the density of states. All of these quantities reflect the constituent elements in the superconducting compound and, as such, are subject to the systematic variation of elemental properties embedded in the periodic table.
The method has been applied to recently discovered hydrides, and this algorithm agrees with the observed transition temperatures. It is necessary to test this method on other superconductors at high and low pressures to determine its applicability and the underlying laws. The main goal of research is to find a general law and predict a material with the desired T c value at any pressure.