There are various types of materials that have different levels of electrical conductivity, and one category is known as superconductors or superconducting materials. Superconducting materials are characterized by their complete lack of electrical resistivity. These materials are highly important due to their wide range of applications in electricity transmission, although they do have certain limitations. The Bardeen–Cooper–Schryver theory and the Ginzburg–Landau theory are two significant theories used to explain the nature of superconducting materials. Of particular interest in this study is the Ginzburg–Landau differential equation, which is considered a vital equation in this field. This equation belongs to a class of nonlinear differential equations. Our research focuses on simulating solutions to the Ginzburg–Landau equation under steady-state conditions. We conducted simulations for several superconducting materials, including aluminum, niobium, lead, tin, niobium germanide, niobium tin, vanadium silicate, lead hexa-molybdenum octa-sulfur, magnesium diboride, uranium triplatinum, potassium, barium copper oxide, yttrium, calcium copper oxide, and barium mercury. We define a new parameter of the superconductor conduction materials, which is the periodic parameter of the superconductor. By analyzing the periodic solutions obtained from the Ginzburg–Landau differential equation, we were able to determine the values of the periodic penetration parameters for each material. Notably, monatomic superconducting materials exhibited periodic penetration parameters in the range of tens of micrometers, while tetra- and penta-elements materials had values in the tens of nanometers. Superconducting materials of two or three different elements showed average values for these parameters. These findings provide valuable insights into the characteristics and behavior of various superconducting materials.

## INTRODUCTION

There are multiple types of materials in terms of their specific electrical conductivity. Conductors are materials that have high specific electrical conductivity, while insulators have low specific electrical conductivity. One significant type of material in terms of specific electrical conductivity is superconductors. Superconducting materials are a class of materials that have zero electrical resistance and extremely valuable electrical conductivity. They can be classified according to the critical temperature of the superconductor or according to the phase transition. There have been numerous studies on superconducting materials, including experimental and theoretical methods. Ginzburg–Landau’s (GL) theory has been one of the most prominent theories in explaining superconductors, and many studies have been conducted based on this theory. For example, Dang *et al.*^{1} made a theoretical prediction about the distribution of vortices in a circular disk with a sector defect. This sector defect acts as a gateway for the intentional entry of vortices into the ring disk, and the researchers found that the number of vortices in the outer shell depended on the number of vortices in the inner shell when the magnetic field was increased uniformly. Superconducting vortices can be used in various applications, such as magnetic resonance imaging, to generate magnetic fields. Panna and Islam^{2} took a new approach to calculating the speed of front propagation from the super region to the normal region in a superconducting sample. Pack *et al.* also studied surface defects in the farthest manifold within the framework of Ginzburg–Landau theory. Type II superconductors are characterized by a phase transition from a pure superconducting state to a mixed state with arrays of magnetic vortices.^{3} The Ginzburg–Landau theory uses two significant parameters: the penetration depth of London (*λ*) and the correlation length of superconducting (*ξ*).^{4} The ratio between these parameters determines the type of the superconductor. Akiyama and Shibata also studied the Lp (Lp Framework) methodology for static and variable Ginzburg–Landau theorem problems.^{5} New methods are being sought to solve the Ginzburg–Landau–Maxwell equations; a set of mathematical equations used to describe the behavior of superconducting materials. The Ginzburg–Landau–Maxwell equations consist of two equations that describe the state of superconducting matter, while the Maxwell equation describes the magnetic field. The approach proposed in this research utilizes a technique called Lp analysis, which allows for the study of the properties of mathematical solutions without the need to know their exact expression. The focus of the research is to solve both stable and unstable equations. Stable equations are those that remain unchanged over time, whereas unstable equations are those that change over time. Chen and Guo conducted a classical study of helium gas near the Bardeen–Cooper–Schrieffer (BCS) junction.^{6} Classical solutions refer to solutions of mathematical equations that do not involve nonlinear movements or oscillations. The research employs a method known as continuity analysis, which enables the study of the continuous behavior of mathematical equations as the values of the mediators change. The research discovered that classical solutions of the time-dependent Ginzburg–Landau equations for atomic helium gases near the BCS-BEC (Bose–Einstein condensation) junction continue to exist as the density approaches the critical density of the junction. In this paper, we obtain classical solutions of the time-dependent Ginzburg–Landau (TDGL) equations originating from superfluid atomic helium gases near the Feshbach resonance of the fermion–boson model. Through the application of Besov and Sobolev spaces, matrix theory, and the energy method, we establish the universal and unique existence of classical solutions of the TDGL equations in the case of the BCS-BEC junction. Here, TDGL equations refer to time-dependent Ginzburg–Landau equations, superfluid atomic Fermi gases refer to superfluid atomic Fermi gases, Feshbach resonance refers to a phenomenon in atomic physics where the energy levels of two different atomic states match when a magnetic field is adjusted, and fermion–boson model refers to a theoretical physics model describing the interaction of fermions and bosons. Besov and Sobolev spaces are mathematical spaces used for studying function properties. In Ref. 7, the behavior of a superconducting plate with a transport current in a magnetic field parallel to its surface was investigated using a numerical solution of the Ginzburg–Landau (GL) equations. Boundary conditions were used for the order coefficient in their general form. These boundary conditions allow for consideration of the effect of the sheet boundaries on the superconducting state inside it. According to the calculations, a dependence of both the critical current and the critical field parallel to the surface of the plate on the sheet’s thickness was discovered. Based on the calculations, an approach was proposed to estimate the correlation length *ξ.* The results of the calculations align with experimental data and qualitative analysis based on the GL theory. Meanwhile, the study^{8} focused on vortex states in small star-shaped sheets of Mo80Ge20. The authors reported on the analysis of the structure of vortices in these sheets, using both theoretical and experimental methods. Theoretical calculations were performed using the nonlinear Ginzburg–Landau theory to generate numerical results for two-dimensional models of star-shaped superconducting sheets, allowing for the study of vortex distribution as a function of vorticity L (number of vortices per unit area) and sample volume. Experimental measurements were conducted using a superconducting quantum interference device (SQUID) to record the distribution of vortices in the star-shaped superconducting sheets and the very small magnetic fields produced by the vortices. In the study,^{9} type II superconductivity was demonstrated in a type of chalcogen zirconia tripyride crystal. The material ZrP1.54S0.46, which was carefully synthesized and tested using a variety of techniques, exhibits almost perfect type II behavior with a Ginzburg–Landau *κ* coefficient of 24. Vacancies in materials can lead to various undesirable properties, including a decrease in strength and electrical conductivity. Recently, scientists have discovered superconducting phenomena in a wide range of iron-based materials, leading to extensive research on compounds corresponding to other transition elements.^{10} For instance, BaNi2P2, a layered material containing NiP layers that facilitate electron transfer, shows superconductivity at a critical temperature *T*_{c}* *= 2.5 K similar to the iron arsenides of the 122 family, suggesting that this nickel-based compound could be another example of unconventional superconductivity. Unconventional superconductivity is a phenomenon that occurs in certain materials that do not adhere to the Migdal–Eliachberg theory, which is based on electron–phonon interaction for Cooper pair formation. The authors’ findings provide valuable insights into the superconducting behavior of BaNi2P2 and other nickel-based compounds, suggesting that these compounds may be promising candidates for new applications of Migdal–Eliashberg superconductors. The existence and stability of weak solutions of the time equations of the Ginzburg–Landau theorem are examined in non-convex three-dimensional curved polygons, where the magnetic field derivative may not be integrable.^{11} In the introduction, the authors discuss the scientific background surrounding the time equations of the Ginzburg–Landau theory, which are utilized to describe superconducting phenomena in superconductors. The non-Abelian Ginzburg–Landau theory of superconductivity provides a framework for describing superconductivity in materials containing non-Abelian charged superconductors.^{12} In non-Abelian superconductivity, electrons are paired together to form Cooper pairs with non-Abelian charge. The properties of superconducting resonators with ultra-thin NbTiN films are examined.^{13} Superconducting resonators are electrical devices that utilize the superconducting properties of superconductors and can be employed for various applications such as sensing, measurement, and signal processing. The extended Ginzburg–Landau equations, Aberikov solution, and the geometric transition from a square to rectangular grid in the magnetic field are discussed.^{14} Aberikov’s solution represents a solution of the time equations of the Ginzburg–Landau theorem in a magnetic field, describing the presence of superconducting vortices in the superconductor. The geometric transition from a square to rectangular lattice refers to a phenomenon where the periodic lattice of supervortices in the superconductor undergoes a shape change. The article explores the unconventional superconducting properties observed in Sr2RuO4.^{15} This material is believed to exhibit a type of superconductivity known as “triplet superconductivity,” where the density of the superfluid can be transferred between different types. The model predicts the existence of two phases of superconductivity in Sr2RuO4, and the theoretical predictions of this model align with available experimental data. In the study,^{16} a comprehensive set of superconducting parameters for the k3C60 superconductor was determined. This study presents a proposed method to find periodic solutions to the Ginzburg–Landau equation in superconductors for a variety of pure and complex superconducting materials. The second section discusses the solution methodology, while the third section presents the results of the solutions for several superconducting materials and the definition of new parameter of the superconductors, which is the periodic penetration parameter. Finally, the most significant conclusions and recommendations are provided.

## METHODOLOGY

*υ*

_{1}, which denotes a positive value, is reliant upon the physical constants. Similarly, υ

_{2}, which represents a negative value, is also contingent upon these same physical constants. The first Landau parameter, denoted as

*α*, and the second Landau parameter, denoted as

*β*, are influenced by the absolute temperature of the superconductor. It is possible to analyze the time-independent form of the Ginzburg–Landau equation by imposing the condition of time independence, which is given as follows:

^{17–20}This method sequentially calculates intermediate values based on the current estimate of the solution, using a weighted sum of these values to update the solution estimate for the next point. By iteratively applying this algorithm, we can approximate the solution of the differential equation at discrete points, while also considering the specific conditions applied to the wave function. The detailed steps of the solution algorithm are as follows:

We impose an initial value for the wave function within the achieved physical conditions.

We choose the appropriate parameters for the superconductor after selecting the considered superconductor.

We write the Runge–Kutta equation for the wave function individually.

The Runge–Kutta equation is written for the first derivative.

The Runge–Kutta equation is written in the Laplacian case.

The rank of the Runge–Kutta method is chosen in finding solutions.

In the pre-final step, the appropriate repetition is applied.

Finally, the step is adjusted to find the solution.

To gain deeper insights into the periodic solutions of the Ginzburg–Landau equation, a meticulous selection process was employed to select multiple superconducting materials for investigation.^{21} The choice of these materials was based on a variety of parameters, aimed at capturing an accurate representation of their unique properties. With the intention of comprehensively covering a broad spectrum of superconductors, our survey encompassed diverse classes including both single elements and compounds. Furthermore, to ensure the integrity of our findings, scans of these selected superconducting materials were conducted at multiple temperatures. This systematic approach allowed us to gather extensive and detailed data, enabling a thorough examination of the periodic solutions within the context of different thermal conditions. By incorporating a robust scientific methodology, our study endeavors to contribute valuable insights into the behavior of superconducting materials under varying circumstances.

## RESULTS AND DISCUSSION

In this section, we will present the most important results that have been achieved in studying superconducting materials using periodic solutions of the Ginzburg–Landau equation. By analyzing the specific periodic solutions, we were able to identify the distances associated with their periodic nature and compare them with the average penetration depth for each material. Our empirical findings demonstrate that the application of the simulation algorithm in diverse superconducting materials ensures the achievement of convergence across all these materials. The convergence of our numerical algorithm aligns with the convergence of the Runge–Kutta method, where the convergence of the Runge–Kutta method pertains to the behavior of the method as the step size (*Δx*) approaches zero. Generally, as the step size (*Δx*) decreases, the accuracy of the approximation improves. The Runge–Kutta method is recognized as a higher-order method, meaning that the error of the approximation decreases at a faster rate compared to lower-order methods. Specifically, the Runge–Kutta method exhibits a convergence order of four, indicating that the error in the approximation is proportional to the fourth power of the step due to the utilization of the fourth order Runge–Kutta. This significant convergence order facilitates the attainment of precise solutions even with relatively large step sizes. However, it is crucial to acknowledge that the efficacy of the method also relies on the attributes of the ODE being solved. Certain ODEs may necessitate smaller step sizes or supplementary techniques to obtain accurate outcomes. Figure 1 depicts the periodic solutions of the Ginzburg–Landau equation applied to aluminum, while Fig. 2 displays the solutions for niobium. The solutions for Pb are presented in Fig. 3, and for tin in Fig. 4.

The periodic solutions we discovered include aluminum, niobium, lead, tin, niobium germanide, niobium tin, vanadium silicate, lead hexa-molybdenum octa-sulfur, magnesium diboride, uranium triplatinum, barium copper oxide, yttrium, calcium copper oxide, and barium mercury.

The previous four materials are examples of pure elements superconducting materials.

Figure 5 shows the periodic solutions of the Ginzburg–Landau equation for magnesium diboride, while Fig. 6 displays the solutions for niobium tripartite germanide. Figure 7 presents the solutions for vanadium trisilicate, and Fig. 8 showcases the solutions for uranium triplatinum.

In terms of the type of atoms, the previous four materials are regarded as an illustration of two types of elements superconducting materials.

We present the solutions of the periodic Ginzburg equation for lead molybdenum hexadecane sulfur in Fig. 9. This compound serves as an illustration of three-elements superconducting compounds based on the included elements types.

Figure 10 displays the solutions of the Ginzburg–Landau equation for yttrium dibarium tricopper oxide, which is an illustration of a superconducting material consisting of four elements.

In Fig. 11, we present the periodic solutions to the Ginzburg–Landau periodic equation for the compound mercury dibarium dicalcium tricopper trioxide, which consists of five elements.

Through our investigation of a variety of superconducting materials that possess distinct compositions, we have uncovered recurring values for the depth parameter within the Ginzburg–Landau equation. Specifically, we have identified a periodic penetration parameter, denoting the periodic solution of each individual superconductor. By meticulously characterizing these periodic parameter values *λφ*, we have compiled the resulting data into three comprehensive tables. Table I encompasses the monoatomic superconductors, Table II encompasses the two types elements superconductors, and Table III encompasses the superconductors composed of three, four, and five types elements. Each table effectively showcases the corresponding values pertinent to their respective superconducting materials.

Superconductor . | λφ (cm)
. |
---|---|

Al | 1.548 × 10^{−3} |

Nb | 7.524 × 10^{−5} |

Pb | 8.826 × 10^{−5} |

Sn | 5.800 × 10^{−4} |

Superconductor . | λφ (cm)
. |
---|---|

Al | 1.548 × 10^{−3} |

Nb | 7.524 × 10^{−5} |

Pb | 8.826 × 10^{−5} |

Sn | 5.800 × 10^{−4} |

Superconductor . | λφ (cm)
. |
---|---|

Nb_{3}Ge | 1.762 × 10^{−7} |

Nb_{3}Sn | 3.390 × 10^{−7} |

V_{3}Si | 4.480 × 10^{−7} |

MgB_{2} | 1.374 × 10^{−7} |

UPt_{3} | 5.423 × 10^{−7} |

Superconductor . | λφ (cm)
. |
---|---|

Nb_{3}Ge | 1.762 × 10^{−7} |

Nb_{3}Sn | 3.390 × 10^{−7} |

V_{3}Si | 4.480 × 10^{−7} |

MgB_{2} | 1.374 × 10^{−7} |

UPt_{3} | 5.423 × 10^{−7} |

Superconductor . | λφ (cm)
. |
---|---|

PbMo_{6}S_{8} | 4.810 × 10^{−8} |

YBa_{2}Cu_{3}O_{7−δ} | 3.590 × 10^{−8} |

HgBa_{2}Ca_{2}Cu_{3}O_{8+δ} | 2.250 × 10^{−8} |

Superconductor . | λφ (cm)
. |
---|---|

PbMo_{6}S_{8} | 4.810 × 10^{−8} |

YBa_{2}Cu_{3}O_{7−δ} | 3.590 × 10^{−8} |

HgBa_{2}Ca_{2}Cu_{3}O_{8+δ} | 2.250 × 10^{−8} |

The presented tables provide a clear visualization of the periodic parameter or periodic depth parameter, showing that aluminum (Table I) has the highest value, while yttrium dibarium tricopper oxide (Table III) has the lowest value. It is evident that pure mono-atomic materials, when compared to other materials, consistently exhibit high parameter values. Across all the mono-atomic materials analyzed, these values reach tens of micrometers. This trend emphasizes the unique characteristic of mono-atomic materials in terms of their periodic parameter.

On the other hand, tetragonal and pentameric materials have remarkably smaller values, in the range of tens of nanometers. Additionally, two types elements superconducting materials (Table II) and three types elements superconducting materials fall between these two categories with average periodic penetration parameter values. The main motivation of this study is to understand and analyze the behavior of different superconducting materials using Ginzburg–Landau’s theory.

The Ginzburg–Landau theory is instrumental in explaining different aspects and phenomena related to superconductors, including the existence of various types of superconducting materials. Our main goal was to analyze and investigate the behavior of several superconducting materials by simulating solutions to the Ginzburg–Landau equation. We selected materials such as aluminum, niobium, lead, tin, niobium germanide, niobium tin, vanadium silicate, lead hexa-molybdenum octa-sulfur, magnesium diboride, uranium triplatinum, potassium barium copper oxide, yttrium, calcium copper oxide, and barium mercury based on their atomic structure and significance as superconducting materials. We examined the obtained solutions to determine the periodic parameter of penetration for each material and observed that single-atomic superconducting materials (Table I) have a larger penetration periodic parameter (in the tens of micrometers range), while tetra- and penta-elements materials have a smaller penetration periodic parameter (approximately in the tens of nanometers range). Two types elements superconducting materials and three types elements superconducting materials exhibit intermediate values between these two categories, as shown in Tables II and III. This indicates that the atomic structure of a material plays a crucial role in its superconducting properties. Superconductivity is the ability of certain materials to conduct electric current with zero resistance. The Ginzburg–Landau theory is a mathematical framework that has been widely used to explain the behavior of superconductors. In our study, we aimed to analyze several superconducting materials by simulating solutions to the Ginzburg–Landau equation. We selected materials like aluminum, niobium, lead, tin, etc., based on their atomic structure and importance as superconducting materials. The obtained solutions provided insights into the behavior of these materials, with the penetration periodic parameter being of particular interest. This parameter represents the distance over which the superconducting properties penetrate into the material. We found that single-atomic superconducting materials have a larger penetration periodic parameter, typically in the micrometer range. In contrast, tetra- and penta-elements materials have a smaller penetration periodic parameter, usually in the tens of nanometers range. The results from Table III demonstrate this distinction. This observation highlights the significant impact of atomic structure on superconducting properties. Materials with a larger penetration periodic parameter are more effective at carrying electric currents over longer distances without resistance compared to those with a smaller penetration periodic parameter. The findings in Tables I–III provide valuable insights into the behavior of superconducting materials and their potential applications. By understanding the relationship between atomic structure and superconducting properties, researchers can explore the development of new materials with enhanced superconductivity characteristics. Through the use of the Ginzburg–Landau theory and our simulations, we have gained valuable knowledge about various superconducting materials. This knowledge can contribute to advancements in superconductivity research and the creation of new technologies that benefit from the unique properties of superconductors.

The aforementioned materials classes—monatomic, two types elements, three types elements, four types elements, and five types elements—serve as exemplary representatives of two distinct categories of superconducting materials, namely, monoatomic superconductors and non-monoatomic superconductors, based on the types of atoms present within. These materials provide an intriguing insight into the behavior and properties displayed by these elemental compositions when they achieve superconductivity. Upon closer examination, it becomes evident that these materials can be classified into two groups, based on the elemental composition. This distinction is crucial in comprehending and studying the diverse range of superconducting behaviors observed in different materials. By further exploring these classifications, a deeper understanding of the unique characteristics exhibited by each group can be achieved. The first category encompasses materials composed of transition metals, such as niobium and tantalum, which demonstrate superconducting properties. These transition-metal based superconductors possess distinctive electronic structures that facilitate superconductivity at low temperatures. The discovery and utilization of these materials have brought about revolutionary advancements in various technological fields, including the development of high-performance magnets and particle accelerators. The second classification includes superconducting materials formed from compounds containing rare earth or alkaline-earth metal elements. Examples of these compounds include magnesium diboride and lanthanum cuprates, which exhibit superconductivity at temperatures significantly higher than traditional elemental superconductors. The unique structural properties and electronic configurations of these compounds contribute to their ability to achieve superconductivity at elevated temperatures. This breakthrough has paved the way for potential applications in electrical power transmission and efficient energy storage.

Expanding our understanding of these two elemental classifications will drive further progress in the field of superconductivity. Scientists and engineers can delve deeper into the intricate relationships between atomic composition, crystal structure, and superconducting properties to develop novel materials with enhanced superconducting capabilities. The pursuit of new superconducting materials, particularly those with elevated critical temperatures, holds immense potential for transforming various technological domains and revolutionizing our energy landscape.

## CONCLUSIONS

The Ginzburg–Landau theory is widely regarded as a significant contribution to the study of superconducting materials because it explains numerous properties and phenomena associated with superconductors. This theory provides a valuable mathematical framework for understanding the behavior of superconductors. Our research focused on simulating solutions to the Ginzburg–Landau equation for various classes of superconducting materials. We selected these materials based on their importance as superconductors and their atomic structures. Our simulations revealed that single-atomic materials have a larger penetration periodic parameter, indicating their ability to conduct electrical currents over long distances without resistance. In contrast, tetra- and penta-elements materials exhibit a smaller penetration periodic parameter, while two types elements superconducting materials and three types elements superconducting materials have intermediate values. This suggests that the atomic structure of a material affects its superconducting properties. Understanding this relationship can aid in the development of new materials with improved superconductivity. Overall, our study contributes to the advancement of superconductivity research and the potential for technological applications. We also defined the periodic penetration parameter of the Ginzburg–Landau equation. Our main objective was to gain a deeper understanding of superconducting materials and their behavior by using Ginzburg–Landau’s theory and analyzing solutions in different materials. We employed numerical simulation methods to examine steady state solutions of the Ginzburg–Landau equation and applied them to various superconducting materials including aluminum, niobium, lead, tin, niobium germanide, niobium tin, vanadium silicate, lead hexa-molybdenum octa-sulfur, magnesium diboride, uranium triplatinum, barium copper oxide, yttrium, calcium copper oxide, and barium mercury. Our selection of materials was based on factors such as their classification within the first or second categories of superconducting materials and their diverse atomic structures. By analyzing these solutions, we calculated the penetration periodic parameter for each superconducting material. Our findings showed that single-atomic superconducting materials had penetration parameters in the range of tens of micrometers, while tetra- and penta-elements materials had parameters estimated at tens of nanometers two types elements superconducting materials and three types elements superconducting materials fell between these extremes. We believe that our study of periodic simulations of superconducting materials using the Ginzburg–Landau theory can be expanded to include other significant superconductors such as gold ormus and other superconductors.

## ACKNOWLEDGMENTS

This research was funded by Damascus University, https://damascusuniversity.edu.sy.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

M.A.A., M.H., and M.A.-R. contributed equally to this work. All authors are marked as first author.

All authors of the manuscript (Marwan Al-Raeei, Mohamad Asem Alkurdi, and Mohamad Hassoun) are responsible on the Conceptualization; Data curation; Formal analysis; Funding acquisition; Investigation; Methodology; Resources; Software; Validation; Visualization; Roles/Writing - original draft; and Writing - review and editing design, writing, methodology, preparing the finale manuscript file. Marwan Al-Raeei is the supervisor of the research and is responsible on Project administration.

**Mohamad Asem Alkourdi**: Conceptualization (equal); Data curation (equal); Formal analysis (equal). **Mohamad Hassoun**: Conceptualization (equal); Data curation (equal); Formal analysis (equal). **Marwan Al-Raeei**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.

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