Circuit-Based Approaches for a Superconducting-Like Behavior with Relatively High Critical Current Density

1. Introduction

Let us consider a brief history of superconductor studies as the background for this paper. Since the initial discovery of superconductivity, more than a century has passed. During this time, many significant achievements have been made. For example, Bardeen–Cooper–Schrieffer theory [1], which found that the carrier of a superconductor has the combination of a Cooper pair, was established. Then, ceramic cuprates whose critical temperatures [2,3,4,5,6] are beyond that of LN₂ were found. Further, MgB₂ [7,8] and Fe-based superconductors [9,10,11,12,13,14] appeared. Moreover, recently H-based superconductors with high critical temperatures and with high pressures [15,16,17] were found. These superconductors promoted studies in condensed matter physics [21,22,23,24,25]. However, the common weak point of the above superconductors is that they require refrigeration or pressures. To overcome this weak point, many previous studies have created some compounds which would be expected to work at higher temperatures. However, most of them did not succeed.

Approximately 10 years ago, we demonstrated a new type of superconductivity using theoretical and experimental techniques [18,19,20]. That is, it is a circuit-based approach not a compound. This superconductivity is generated when a diffusion current from a current source is supplied to a doped semiconductor and an electrostatic field from a condenser cancels the Ohmic voltage of the semiconductor. As a result, the internal voltages are zero but the current remains without refrigeration. Accordingly, this new superconductivity bypasses the problem of refrigeration because it is not related to temperatures.

In the present paper, we describe further new type of superconductivity, which demonstrates extreme progress compared with the above-mentioned superconductivity discovered 10 years ago. The features of this study are

• The method generates superconductivity without refrigeration and without pressures;
• Its critical current is sufficiently high;
• It does not require specific substances or the mounting of specific setups; and
• The mechanism of this new type of superconductivity is explained in terms of condensed matter physics.

To distinguish the abovementioned two new types of superconductivity, we will refer to the one developed 10 years ago as PNS (referring to the Previous New Superconductivity).

Compared with PNS, the present superconductivity differs in the following aspects.

• The critical current density is much larger. This fact is important when considering practical applications. In PNS, the critical current was less than 10 μA, which prevents it from being used in practical applications; however, the critical current of the present superconductivity is estimated to be 10⁴ A order. This value will pose no problems for practical applications.
• It is clear that the mechanism is the Meissner effect. In PNS, we knew that the superconductor discharged a current as a result of the application of static magnetic fields. However, the details of that mechanism were not clear. In the present study, producing a new electric field, we were able to derive the concrete London equation and identify the mechanism as the Meissner effect. A key point is that the sample in an experiment discharges a current to release the additional energy derived from the applied magnetic field energy.
• Numerical simulations result in clear superconductivity. In PNS, we could not establish a simulation method; however, the present superconductivity system implies a pure electrical circuit, which enabled us to employ simulation software for electrical circuit calculations (e.g., the PSIM software) by introducing an equivalent circuit. The use of this simulation method enabled us to investigate this system via various approaches.
• It is not necessary to prepare the specific device. In PNS, we had to produce the device which is a sandwich of semiconductor, insulator and condenser pole plates. Although this device confirmed the zero resistance, it is hard to put it into general practice. However, the present method of this paper does not require the production of the device or materials. This is a new and significant point compared with PNS.

To conclude, the mechanism of our new superconductivity including PNS enables a comprehensive understanding of the generated superconductivity.

As relevant fields, superconducting circuits, quantum circuits or micro-circuits are studied by another researcher [26,27,28,29]. However, most of them employ the existing refrigerated superconductors. For example, the concepts of the existing superconductors are employed like Josephson junctions, a critical temperature, a critical current or an order parameter and so on. However, our superconductivity does not require any refrigeration, which is different from the existing superconductors.

In this paper, we will first review the characteristics of both the voltage and current sources. Then, the principle of our system is described. The mechanism of pairing and forming the macroscopic wave function is discussed theoretically with obtaining the London equation. Moreover, a critical current equation is formulated. The Methods section proposes an equivalent circuit with calculations of the inductance of the internal toroid. In the Results section, we calculate concrete values of the critical currents and use the PSIM software to calculate the time dependences of the voltages and currents in the sample.

Moreover, importantly, we have prepared Appendix A of a guide to reproduce the experiments and of the presentation of preliminary experimental results. We believe that this appendix is significant because it provides a solution of obtaining a superconductor, which enables us to enhance performances of every electrical device. Although it is all right that the results of the experiments in this appendix are placed in the main body but because they are preliminary tests, we located the descriptions at the end of the paper as an Appendix A.

In summary, the contents of the theory are simple but novel and thus it provides a new knowledge to the field of condensed matter physics like a new paring system, new forces or a new type of Bose-Einstein condensation. Furthermore, this theory is supported by the numerical results and the preliminary experimental results from Appendix A.

2. Principle

2.1. Review of Voltage and Current Sources

In our proposed system, voltage and current sources are employed, which are elements of a general electrical circuit. Figure 1 shows schematics of these sources. The voltage source generates a constant voltage while its current is varied depending on the connected load. The internal resistance of this voltage source is ideally zero. Conversely, the current source supplies a constant current (not voltage) to the load, which is not related to the electrical resistance of the load. Therefore, its voltage is varied depending on the resistance of the load. In contrast to the voltage source, the internal resistance of the current source is ideally infinite. Moreover, a current from the current source stems from a diffusion current, which is related to a collector current in transistors. This fact will be important later when discussing the theory.

2.2. Principle

As shown in Figure 2, a voltage source, a current source, and a load are connected in series. As previously mentioned, the internal resistance of the voltage source is zero, whereas the internal resistance of the current source is infinite. Moreover, the output voltage is equal to the voltage from the load, as derived via Ohm’s law. Considering these points, the following can be said:

• The generating current from the voltage source is zero because of the infinite resistance of the current source. Therefore, this source generates only the voltage V.
• For the current source, because of the balance of the two voltages of the voltage source and the load, the voltage between the taps of the current source becomes zero. Therefore, this source supplies only the current I to the load.

These characteristics are listed in Table 1.

Considering the above, we see that neither source generates electric power. Therefore, energy conservation requires that the load not receive any energy and that the load not generate Joule heating. Because the current I exists in the load because of the current source, the absence of Joule heating by the load results in zero electric resistance. Therefore, we can predict that this system, in principle, will result in a new type of superconductivity. However, before we reach this conclusion, it is necessary to theoretically examine the mechanism of the superconductivity in terms of condensed matter physics and demonstrate that the Meissner effect is generated.

3. Theory

3.1. Transient State

3.1.1. Spatial Electron Concentration at Transient State

First, we assume that the voltage of the voltage source and the voltage of the load are equal. Then, the diffusion current of the current source is introduced. Considering the conductivity, introducing an average electric field, and substituting a diffusion constant result in a specific and special electron concentration. The voltage balance is

$$V_E=V=RI,$$

(1-1)

$$R=\mathsf{\rho }\frac{l}{S},$$

(1-2)

where V_E, V, R, and I denote the voltage from the voltage source, the voltage of the load, the resistance of the load, and the current from the current source, respectively. In Eq. (1-2), ρ, S, and l denote the resistivity, the area of the load, and the length of the load, respectively. The diffusion current I is expressed as [18,30]

$$I=SqD\frac{dn}{d\xi },$$

(2)

where q, D, and n denote the electron charge, the diffusion constant, and the electron concentration, respectively. Therefore, the Ohmic voltage becomes

$$V=\mathsf{\rho }\frac{l}{S}SqD\frac{dn}{d\xi }=\frac{l}{\sigma }qD\frac{dn}{d\xi },$$

(3)

where σ denotes the conductivity of the load. Here the following equations are substituted:

$$\sigma =qn\mu ,$$

(4-1)

$$D=\frac{\mu k_{B}T}{q},$$

(4-2)

$$V=E_{i}l,$$

(4-2)

where μ, k_B, T, and E_i denote the average mobility, the Boltzmann constant, the temperature, and the local electric field, respectively. Therefore, the local electric field is

$$E_i=\frac{k_{B}T}{qn}\frac{dn}{d\xi }.$$

(5)

Solving this equation gives

$$n=n_0\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{q{E}_i}{k_{B}T}\xi \right).$$

(6)

3.1.2. Attractive Potential and Coiled New Electric Field at the Transient State

We employed the Poisson equation to derive the interaction potential between two electrons at the transient state. Equation (6) gives the concentration in terms of the electrons; substituting this concentration into the Poisson equation produces the following attractive interaction potential ($V_{\alpha }<0)$:

$$V_{\alpha }=-\frac{n_0{k}_B^2{T}^2}{\epsilon E_i^2}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{q{E}_i}{k_{B}T}\xi \right),$$

(7)

where ξ is variable and relative macroscopic distance between the two electrons. Hereafter, we refer to this potential as the transient state potential. It is very important to note that the Coulomb repulsive interaction is also determined using Poisson’s equation. Accordingly, this repulsive interaction does not appear at a macroscopic scale. Instead, the abovementioned interactive potential V_α works. Note that, at the steady state, the macroscopic relative distance ξ becomes zero but microscopic one is not zero. In this sense, at the steady state, we will consider the Schrodinger equation.

In the process of solving the Poisson equation, we derived the following new electric field at the transient state:

$$V_{\alpha }=-\frac{n_0{k}_B^2{T}^2}{\epsilon E_i^2}\mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{q{E}_i}{k_{B}T}\xi \right).$$

(8)

When defining the circle integral of eq. (8) (i.e., a voltage), this integral must end up within a material. This implies that, in the material, a coil voltage must be defined. The spatial integral of Eq. (8) for ξ results in a negative voltage. As discussed later, we will see that, immediately prior to the transition from the normal state to the superconducting state, this negative voltage appears in our simulation. This new electric field appears at the transient state and thus an inductor voltage will be essential. That is, as discussed later, the spatial integral of eq. (8) will form a coil.

3.1.3. Combination of Two Electrons and Critical Current Density

The transient state attractive interaction potential V_α results in two electrons that approach each other until their distance is only the lattice diameter due to the lack of the Coulomb repulsive interaction at the macroscopic scale. At this moment, the macroscopic relative distance ξ becomes zero. Note that, however, microscopic one is not zero. Therefore, the transient state up to the combination of the two electrons is a macroscopic phenomenon. However, only when the potential V_α vanishes do we need to consider the microscopic scale. As described later, however, at this moment, when the potential V_α vanishes, the relative kinetic energy also vanishes. Therefore, we do not need to consider the Hamiltonian of the quantum mechanics only before the combination of two electrons. Instead, it is necessary to consider the Hamiltonian for quantum mechanics for the center-of mass motion as described later.

As a result of the transient interaction V_α, the two electrons take the location of a lattice, and at a lattice (i.e., macroscopic variable ξ = 0), the total energy E_T of the system is expressed as follows:

$$\begin{gathered} E_T=\left|V_{\alpha }\left(\xi =0\right)\right|+V+V_p, \text{and} \\ {V}_{\alpha }\left(\xi =0\right)+P=0. \end{gathered}$$

(9)

Here, V is the spin magnetic potential [18] expressed as

$$V=-\frac{q^2{ħ}^2}{16\pi m^2{\left|z_m\right|}^3},$$

(10)

where m and z_m denote the electron mass and the microscopic relative coordinate, respectively. In Eq. (9), P and V_p are the kinetic and zero-point energies in terms of the Debye temperature at the lattices, respectively. V_p is expressed as follows:

$$V_p=\frac{1}{2}ħ{\omega }_D={\frac{1}{2}k}_B{\theta }_D,$$

(11)

where ω_D and θ_D indicate the Debye angular frequency and temperature, respectively. The following conditions should be satisfied for the two-electron combination:

$$\left|V_{\alpha }(\xi =0)\right|\ge V_p,$$

(12-1)

$$\left|V\right|\ge P.$$

(12-2)

For Eq. (12-1) to be satisfied, both electrons must be located at the zero point of a lattice. This condition, i.e., critical current density, will be considered later. To satisfy Eq. (12-2), the microscopic relative distance z_m (i.e., the coherence) should be

$$z_m\le 1.0\times {10}^{-9}\mathrm{m}.$$

(13)

If Eq. (12-1) is satisfied, then, Eqs. (12-2) and (13) are also satisfied because the zero-point energy implies the quantum fluctuation energy of a lattice. Therefore, when current density is less than the critical current density, only the spin magnetic potential V remains, and the entire energy takes on a negative value, which implies that the two electrons have a net combination, in which a collision between the two electrons becomes completely inelastic. Consequently, these two electrons combine to form a pair.

Even though the kinetic energy in terms of the relative motions exists in the transient state, as mentioned, in the steady state, only the spin attractive force V remains:

$$V=-\frac{q^2{ħ}^2}{16\pi m^2{a}^3},$$

(10-2)

where a denotes the constant coherence of an electron pair.

Therefore, this potential is taken to be the superconducting energy gap of our superconductivity. As mentioned, the two-electron pair combines at the distance of a lattice, which is typically estimated to be 1 Å, and, therefore, the magnitude of the energy gap is estimated to be approximately 10^-18 J. The typical room temperature energy k_BT (where the temperature T = 300 K) is approximately on the order of 10⁻²¹ J. Therefore, this superconducting energy gap is much larger than that of room temperature. This implies that the individual electron pair will not be destroyed via normal heating.

Next, let us consider an equation of the critical current density.

At the critical point, i.e., in the case where the inequality of Eq. (12-1) is maximum,

$$\frac{n_0{k}_B^2{T}^2}{\epsilon E_i^2}=\frac{1}{2}{k}_B{\theta }_D.$$

(15)

From this equation, the concentration n₀ can be expressed as

$$n_0=\frac{\epsilon E_i^2}{k_B{T}^2}\frac{1}{2}{\theta }_D.$$

(15)

In Eq. (14), considering the second equation of Eq. (9), the left-hand side of Eq. (14) implies a kinetic energy P. Therefore,

$$\frac{n_0{k}_B^2{T}^2}{\epsilon E_i^2}=\frac{1}{2}m{v}^2,$$

(16)

where m and v denote the mass of the electron and the velocity in terms of the relative motions, respectively. Substituting Eq. (15),

$$\frac{\epsilon E_i^2}{k_B{T}^2}\frac{1}{2}{\theta }_D\frac{k_B^2{T}^2}{\epsilon E_i^2}=\frac{1}{2}m{v}^2.$$

(17)

That is,

$$\frac{1}{2}{k}_B{\theta }_D=\frac{1}{2}m{v}^2.$$

(18)

Therefore, the relative velocity is derived to be

$$v=\sqrt{\frac{k_B{\theta }_D}{m}}.$$

(19)

Consequently, the critical current density equation is obtained such that

$$j_c=q{n}_{0}v=q\frac{\epsilon E_i^2}{k_B{T}^2}\frac{1}{2}{\theta }_D\sqrt{\frac{k_B{\theta }_D}{m}}.$$

(20)

Note that, in the Results section, we will calculate numerical values of the critical current density using Eq. (20).

3.2. Macroscopic Wave Function and the London Equation at the Steady State

Given that a two-electron pair binds strongly, from the discussion in the previous sections, we now precede to a discussion of Bose–Einstein condensation and the macroscopic wave function at the steady state.

As described later, the Hamiltonian equation of an electron pair in terms of the center-of-mass motion is expressed simply as

$$H=-\frac{{ħ}^2}{2\left(2m\right)}\frac{d^2}{d{x}^2}+U.$$

(21)

The potential U in the above equation indirectly implies the electrostatic potential of the two electrons, i.e., the internal electric field at a lattice:

$$U=-{\int }_{x_1}^{x_2}2q{E}_{i}dx=-2q{E}_i(x_2-x_1),$$

(22)

where x₁ and x₂ denote the positions of the two electrons.

In the steady state, we are now considering the combined electron pair at a lattice, without collisions (i.e., the interactions) with another pair, and thus this temperature-independence allows us to consider that the dimension of the center-of-mass motion whose direction is along E_i (i.e., the movement depends on eq. (22)) is single. Note that the distribution of internal electric field E_i at a lattice is also single dimension. The reason of the absence of interactions between pairs and pairs will be discussed soon. However, at the transient state, the existence of the potential V_α implies that each electron motion depends on temperature T. Moreover, because there are not net collisions between pairs and pairs at the steady state, it is potential for a phase transition that is related to temperatures not to exist [25]. The absence of the interactions with another pair guarantees the physical picture of one body.

The local placements at the identical lattice of the two electrons (i.e., the local and strong combination of the two electrons at a lattice) imply that x₁ = x₂. Employing the Hamiltonian equation, a wave function of the pair in terms of the center-of-mass motion can be obtained. Note that, at the moment when the electrons have an identical location at a lattice, the potential U in the Hamiltonian equation converts to an eigenvalue (i.e., the kinetic energy in terms of the center-of-mass motion) that stems from the local electric field E_i. Accordingly, the distribution of the local electric field E_i within the lattice vanishes.

$${\phi }_i=\gamma \exp \left[\frac{2m{\mu }_i{E}_i}{ħ}jx\right],$$

(23)

where μ_i denotes the mobility of the pair at the lattice. Using the following equations,

$$\frac{J}{{\sigma }_i}=E_I \text{and}$$

(24-1)

$${\sigma }_i=\frac{2q}{△V_i/i}{\mu }_i,$$

(24-2)

The wave function of an electron pair can be derived such that

$${\phi }_i=\gamma \exp \left[\frac{2mj}{ħ}xJ\frac{△V_i/i}{2q}\right]$$

(25)

where σ_i, △V_i, and J denote the conductivity at the lattice, the variation quantity of the volume with respect to the increase in the index i, and the current density, respectively.

Equation (24-1) implies the proportion of Joule electric and electrostatic fields at the lattice-scale level. As mentioned, the above wave function of an electron pair is indexed by i, and a wave function at a subsequent lattice does not interact with this wave function because the distance between the two neighboring lattices is much larger than the coherence of the two electrons. Note that, in the previous section, we mentioned that the coherence of a combined pair is estimated to be approximately 1 Å (i.e, the typical diameter of a lattice). This is the reason why each electron pair does not interact with each other.

The entire wave function is represented in the following equation:

$$\mathsf{\psi }=\prod {\phi }_i={\gamma }^i\exp \left[\frac{2mj}{ħ}x\sum J\frac{\frac{\Delta v_i}{i}}{2q}\right].$$

(26)

Assuming that the index i is infinite and that the sample volume of a load is kept constant, the lattice volume △V_i /i converges to the differential dV. Accordingly, the countable lattice concept becomes ineffective, and a macroscopic continuous body appears. That is, we obtain a macroscopic wave function that is mandatory when considering the mechanism of the new superconductivity, and this macroscopic wave function is needed to derive a London equation (i.e., a Meissner effect):

$$\to \mathsf{\eta }\exp \left[\frac{2mj}{ħ}x\int \frac{J}{2q}dV]=\eta \exp [\frac{2mj}{ħ}xa\int \frac{J}{2q}dS\right]=\eta \exp \left[\frac{mI}{ħq}ajx\right].$$

(27)

In Eq. (27), the wave number K is

$$K=\frac{mI}{ħq}a=\frac{I}{2M}a,$$

(28-1)

where M denotes the spin magnetic moment of an electron and a is the constant coherence in terms of our new superconductivity. The spin magnetic moment in this equation can be correlated to the combination energy (i.e., the spin interaction) from Eq. (10) when the microscopic ordinate z_m is replaced with the constant coherence a.

$$K=\frac{I}{2\sqrt{-4\pi aV}}\hspace{1em}\hspace{1em}V\le 0,$$

(28-2)

where V denotes the spin interaction given in Eq. (10-2).

In Eq. (28-2), the macroscopic current I is independent of the collision time and is therefore a superconducting current. Further, the wave number K in this equation is uniform and is not related to space, e.g., it is not related to the lattice index i. Therefore, all the quantum states converge to a single state, and Bose–Einstein condensation is derived.

Let us consider the Meissner effect derived from the macroscopic wave function. As mentioned, the macroscopic wave function is

$$\psi =\eta \exp \left(\frac{mIa}{ħq}jx\right),$$

(27)

In this equation, the macroscopic current I appears. In general, a current needs to be continuous in space and time. In this case, therefore, the current I is continuous for the output current I from the current source. Accordingly, this superconducting current must generate a self-magnetic field in the load sample and, if a new superconductivity is generated, this self-magnetic field must be canceled out.

Considering the above self-magnetic field A, the Aharonov–Bohm (AB) effect [11] can be introduced to the macroscopic wave function, Eq. (27):

$${\psi }_A=\left|\psi \right|\mathrm{e}\mathrm{x}\mathrm{p}\left[\right(\frac{mIa}{ħq}x+\frac{q}{ħ}\int Ads\left)j\right].$$

(28)

Based on the property of phases,

$$\frac{mIa}{ħq}x+\frac{q}{ħ}\int Ads=2n\pi .$$

(29)

Taking the differentials to both sides for variable x,

$$\frac{mIa}{ħq}=-\frac{q}{ħ}\frac{d}{dx}\int Ads.$$

(30)

That is,

$$\frac{mIa}{ħq}=-\frac{q}{ħ}A\frac{ds}{dx}.$$

(31)

$$I\equiv \int j_{n}dS.$$

(32)

Introducing a small angle parameter φ to the differential at the right side,

$$\frac{ma}{ħq}\int j_{n}dS=-\frac{q}{ħ}A\phi .$$

(33)

Assuming a differential for integrals for the surface is very small, Eq. (33) becomes

$$\frac{ma}{ħq}{j}_n\times a^2=-\frac{q}{ħ}A\times \phi .$$

(34)

Therefore, we derive a London equation:

$$j_n=-\frac{q^2\phi }{m{a}^3}A.$$

(35)

3.3. Consideration of the Meissner Effect

When considering a sample with a cylindrical shape, it is necessary to introduce an internal toroid whose large radius corresponds to that of the cylindrical sample. Then, we consider the renewable coordinate that is tangential to the large circumference as corresponding to the z-axis in cylindrical coordinates; see Figure 3. We refer to this type of coordinates as the “specific cylindrical coordinates.”

From the initial time to the transition time t_c, there is a normal self-magnetic field distribution B₀ in the sample along the direction of the z-axis of the specific cylindrical coordinates. However, due to the existence of the new electric field eq. (8) at the transient state, internal voltages must be defined within a material. This implies that coils appear at the transient state within the material. In the process of forming coils, negative voltage appears and currents are produced. Owing to the continuous property of the current, this generated current becomes the persistent one of eq. (35) at the steady state. In this process, the generated magnetic field cancels the existing self-magnetic field B₀ whose direction is along the z-axis of the specific cylindrical coordinates. Therefore, the internal net magnetic flux density becomes zero despite the existence of the current I. That is, from energy conservation, the magnetic field energy derived from B₀ is transformed as a discharged current, which generates a temporal negative voltage, as mentioned previously.

Let us consider the above more concretely. Here, in the right panel of Figure 3, we consider one element coil. However, as discussed soon, this element coil is divided into many local coils. Assuming that the length l in the left panel of Figure 3 is considered to be Na, where N and a denote arbitrary integer and the coherence of a pair respectively, in the conductor, many local divided coils and currents exist. However, considering the analogy of the Stokes’ theorem, these internal local currents of each local divided coil cancel with each other and the net current eventually becomes the most outer current of the blue surface in Figure 3. These phenomena imply that, as described in the theory section, the cancelation of the local currents indicate the perfect inelastic collisions of two electrons and the two electrons bind by the spin magnetic force eq. (10). After that, these combined electrons move uniformly by the local electric field E_i and thus they result in eq. (27) (i.e., the Bose-Einstein condensation). However, the superconducting current I in eq. (27) still generates a self-magnetic field B₀ and thus AB effect leads to the London equation (35) whose current distributes along the most outer line of the blue surface in Figure3. Note that, however, forming of BE condensation and London equation actually occur simultaneously. Thus, the exclusion of the internal magnetic field B₀ at the transient state is taken over to the steady state. Note that, importantly, if the further magnetic field is applied after the steady state, in order to maintain the value of the persistent current of eq. (35), a current is also discharged.

4. Method

4.1. Equivalent Circuit

In the previous section, it was found that an internal, local and divided toroid inductance L is produced in the sample. Considering this, we can introduce an equivalent circuit for our system, shown in Figure 4, which has an inductance L.

The sample has both a resistance R named “R1” and an inductance L named “L2”. Moreover, generally any substance has a small inductor factor (i.e. a flux) when subjected to the transport-current. Thus, in the equivalent circuit, this inductor factor L₀, which is connected by direct connection with “R1”, is employed. This small inductor factor generates a self-magnetic field. Note that these inductances do not give any influences to the voltage balance shown in Figure 2, because Figure 2 implies the steady state and thus sufficiently long time makes the influences of the inductances cease, and we will forecast that the superconductivity appears.

For the numerical calculations, we employed the PSIM electrical circuit software, which can be purchased from their website.

4.2. Calculation of Inductance L

To simulate the equivalent circuit, we need to estimate a concrete value of the inductance L named “L2”, which is determined by the geometrical factors of a local divided coil. As mentioned, it is necessary to consider the specific cylindrical coordinates, i.e., an internal toroid. The inductance of the toroid is given as

$$L=\frac{{\mu }_0{N}_0^{2}S}{{\left(2\pi R\right)}^2},$$

(36)

where μ₀, N₀, S, and R denote the magnetic permeability in a vacuum, the turn number, the area, and the radius of the toroid, respectively. Note that the area S is indicated in Figure 3 as a blue surface. However, L2 indicates the transient coil and thus we now consider each divided local coil. Note that each divided local coil is independent to each other until two electrons combine and their currents are canceled. That is, although at the initial state each dived local coil coexists within the blue surface in Figure 3, they are not the series connections or the parallel connections. Thus, the behaviors of the local coils can be represented by an inductance of one local coil. Note that, considering voltage-current characteristics of an inductor, although the voltage and current vary with time, that inductance still remains. For details, see Section 3.3. In eq. (36), the turn number is derived from the fact that each element coil of the internal toroid is distributed with a relative distance that is equal to the coherence a of our superconductivity (see the right panel in Figure 3):

$$N_0\approx \frac{2\pi R}{a},$$

(37)

where the coherence a is approximated as 1 Å and thus eq. (37) becomes large. Assuming that $S\equiv Ra,$ where a is coherence and R is radius for the cross section for the current and that $R\equiv 1\times {10}^{-5}m$, for example, the inductance L is calculated as

$$L\approx 0.12 \mathrm{H}.$$

(38)

4.3. Resistance of a Load

In general, the value of the resistance of a load is determined by both the conductivity and its shape. In this paper, under the condition that the shape of the load of the sample is kept constant, the conductivity is varied. That is, while the inductance L named “L2” is kept constant, the resistance named “R1”of the sample is varied.

5. Results and Discussion

In this paper, to overcome the weak points of the refrigeration and of adding high pressures for existing superconductors, a temperature-independent superconductivity was presented. Here let us consider the results when operating our system. Note that variations of the value of resistances will confirm the reproductions of the phenomenon.

5.1. Circuit Simulations

First, we show the result when the load is 3 Ω and the input current is 0.5 A. Figure 5a–c indicate the electrical potentials according to Figure 4. Figure 6 shows the time dependence of each current that exists in the sample. Note that sampling time is 0.01 s.

From the initial time, the probe V4 detects very large negative electric potentials up to 0.2 s. However, at t = 0.2 s, V5 becomes nearly zero and V4 also becomes zero. In addition, in Figure 6, the resistance R1 current reaches zero at 0.2 s, which implies the Joule heating becomes zero and inductor L2 current reaches the steady value at 0.2 s. This implies that the tap voltage between V5 and V4, i.e., the voltage of a sample, becomes zero but the current is retained. As mentioned, a large negative voltage occurs immediately prior to the time of the transition. The negative value originated from V4 implies a demonstration of the Meissner effect.

In Figure 7a–c, each electrical potential in the case with 1 Ω is presented, whereas Figure 8 shows the characteristics of the currents in the sample. The reason for varying the resistance is for confirming a reproduced discharged current that corresponds to the additional magnetic field energy (i.e., the Meissner effect). The behaviors of each electrical potential and the current are the same as in the case with 3 Ω; however, the transition time becomes longer shifting to 0.4 s. As an important result, V4 again exhibits a large negative value similar to that in the case with 3 Ω. Therefore, we can conclude that negative voltages appear immediately prior to the transition time demonstrating the Meissner effect predicted by our theory.

Note that each result varies gradually according to time. Thus, as described in the theory section, these results do not define a strict phase transition [25].

Let us consider the result of the probe V3. At the transition, while the V5 reaches 0.0 [V], the V3 detect the output of the voltage source. Considering that the difference between V4 and the V5 at the steady state implies the zero voltage of the sample, only the V3 detect the output of the voltage source. This fact implies that, although the output of the voltage source does not die, the sample voltage (i.e., the sample resistance) reaches zero.

5.2. Values of Critical Current and Comparison with That in PNS

Using Eq. (20), we calculated critical currents. First, let us consider the case of the present paper. When the voltage V = 1 V , then the local electric field is

$$E_i=\frac{V_E}{a}\approx 1.0\times {10}^{10} \mathrm{V}/\mathrm{m}$$

(39)

where V_E and a denote the voltage from the voltage source and the coherence of a pair, respectively. As a result of calculation from eq. (20), the critical current I_c is calculated such that

$$I_c=8.4\times {10}^4\mathrm{A}.$$

(40)

In this calculation, the area S₀ of the cross section for the current was assumed as, for example,

$$S_0\equiv 1.0\times {10}^{-10}{m}^2.$$

(41)

Eqs. (40) and (41) imply that, even though a very thin conductor is employed, a sufficiently large current supply is allowed. The value of Eq. (40) is sufficiently large; therefore, we need not to worry about the basic limitation of the transport current. Table 2 lists the physical constants used in these calculations.

PNS [18,19,20] has the same equation, Eq. (20), for the critical current density. In this case, however, the average electric fields do not stem from the voltage from the output of a voltage source but from the electrostatic fields generated by a condenser. Therefore, E_i in Eq. (20) is derived from the surface charge density σ, which arises from the charge Q on the condenser pole plates. However, because of the extremely small capacitance C (approximately 10⁻¹⁴ F), even when a relatively large voltage is applied to the condenser pole plates, the charge Q is extremely small; in addition, the area of the pole plates is large. Therefore, the surface charge density σ becomes very small. Considering our previous setup [18,19,20],

$$E_i=\frac{\sigma }{\epsilon }\approx 0.11\mathrm{V}/\mathrm{m}.$$

(42)

Considering eq. (20) and the above small internal electric field E_i, the critical current must be small.

That is, the critical current in this case is

$$I_{c,PNS}=2.8\times {10}^{-5}\mathrm{A}.$$

(43)

In Table 3, we indicate the additional constants used to calculate the above value.

Comparing the two critical currents, it is clear that the case proposed in this paper has sufficient technical merit. This is important when considering superconductivity applications.

5.4. Summary to Achieve Our Superconductivity

First, the balance of two voltages and a diffusion current result in a specific electron concentration distribution. Using this distribution and the Poisson equation, a new electric field and a transient attractive potential are generated. Because of this solution of the Poisson equation, the Coulomb repulsive potential as the solution of the Poisson equation does not appear at the macroscopic scale, which results in the extreme short-range approach of the two electrons to each other. Under the condition that the kinetic energy in terms of the relative motions dominates the lattice energy (which involves the concept of the critical current density), the two electrons combine via the spin magnetic potential. This spin magnetic potential implies a superconducting energy gap that is much larger than normal heating at room temperature due to the extremely short coherence. Therefore, the combination of two electrons is not destroyed by normal heating. Given that the two electrons bind very strongly, we consider the center-of-mass motion of an electron pair. As a result, the entire wave function of the center-of-mass motion converges to a macroscopic wave function having a single phase. Using the macroscopic wave function and AB effect, the London equation (i.e., the Meissner effect) was derived.

5.5. Limit of This Superconductivity

To function this system of the superconductivity, a current source properly must work. In other words, the initial input impedance of a sample must be sufficiently small such that the current from the current source can be input. In this sense, a sample having too large input impedance should be avoided to employ.

5.6. Summary of Significances of This Study

Let us review significances of our proposed superconductivity. Thus far, the history of the existing superconductors was explained by considering the critical temperatures. That is, the research approach was previously to create a compound which was expected to work at higher temperatures. However, presently, this approach has not yet solved the problem that a superconductor requires significant refrigeration or pressures. On the other hand, because our circuit-based superconductivity is independent for temperatures, we believe that a new history of superconductors will open.

• 1. We need not prepare specific substances or setups.

It is not necessary to prepare specific substances or compounds because, if the system is implemented correctly, the load is rendered superconductive. Moreover, in PNS, we needed to prepare a setup, in which a condenser, semiconductor disk, and current leads were included. In particular, attaching the current leads to the semiconductor disk was difficult. In the present system, however, such a setup is not necessary, and therefore, we can very simply obtain superconductivity.

• 2. The critical current density is sufficiently high.

PNS has an extremely small critical current [18,19,20], and this fact prevented it from being used in practical applications. In the present superconductivity, however, the critical current is sufficiently high; therefore, we need not be overly concerned by the values of the transport currents.

• 3. It is not necessary to secure extremely low temperatures and high pressures.

To implement our system, it is not necessary to secure extremely low temperatures or high pressures because the critical current density of our superconductivity is sufficiently large and the temperature is not basically related to the generation of the superconductivity.

• 4. Global energy problems might be solved, and some technologies are redirected.

Using our proposed system, efficient energy transmissions or performance improvements of electrical devices like motors, electric wires or magnets can be conducted. Moreover, as demonstrated in the later Appendix A at the end of this chapter, it is possible to make a metal be superconductive, which implies that both the running and manufacturing costs will be significantly reduced. This fact is important to the field of developing electrical devices.

5.7. Important Considerations When Implementing a Superconductor Using This System

Let us consider the time at which the sample load is disconnected from the system after confirming superconductivity. In this case, does the superconductivity remain? We claim that the equivalent circuit in the sample should remain with the persistent current I of the internal toroid. For the details, please see results of the Appendix A in this chapter.

6. Conclusions

In this paper, we proposed a new type of superconductivity, in which circuit-approached and temperature-independent properties exist.

Generally, the existing superconductors require the significant refrigeration or high pressures, which prevent them into practical applications. Nonetheless, the zero-resistance property of the superconductors is very important to every electrical device to improve the performances. Receiving this fact, the present paper proposed a new principle to generate superconductivity with no refrigeration and no pressures.

As a result of both analytical and numerical calculations, a significantly large critical current density and the Meissner effect, as well as the fact that the sample resistance becomes zero, were found. The significance of our superconductivity is that it can be generated very simply with no refrigeration and with no preparation of specific substances or setups and that it achieves a large critical current density. Moreover, this superconductivity does not require high pressures. These properties have not yet been achieved in conventional superconductor studies. As mentioned, however, the presented system cannot make too large load be superconductive.

As a natural follow-up, we need to confirm this phenomenon in actual experiments. However, the Appendix A in this paper presents preliminary experimental results, which confirms the zero resistance and the Meissner effects.