The Critical Magnetic Field and Current Density of an Entangled Yttrium Barium Copper Oxide Superconductor Using Ginzburg‐Landau Theory

1. Introduction

In 1911, Dutch physicist Heike Kamerlingh Onnes discovered that the resistivity of mercury abruptly disappeared at around 4.2 K, the boiling point of liquid helium, when studying the resistance of solid mercury at very low temperatures (Onnes, 1911). In 1933, Meissner and Oschenfeld observed that magnetic flux was excluded from the interior of a superconductor when it was cooled below its critical temperature (Tc) in the presence of an external magnetic field, demonstrating that superconducting materials act as perfect diamagnets (Meissner & Oschenfeld, 1933).

Various theories have been developed to explain superconductivity. The phenomenological theory by Fritz and Heinz London (1935) proposed equations to describe the electromagnetic properties of superconductors using coherence length and magnetic penetration depth to find critical current density and critical magnetic field (London & London, 1935). The semi-phenomenological Ginzburg-Landau theory (1950) and the microscopic Bardeen-Cooper-Schrieffer (BCS) theory (1957) further expanded these ideas (Ginzburg & Landau, 1950; Bardeen, Cooper, & Schrieffer, 1957). For instance, T. Desta et al. (2015) used Ginzburg-Landau theory to investigate the critical magnetic field of HoMo6​Se8​, finding that the critical magnetic field is inversely proportional to the temperature of the superconductor (Desta et al., 2015). Similarly, G. Kahsay et al. (2020) applied Ginzburg-Landau theory to study the upper critical fields of a heavy fermion superconductor (Kahsay et al., 2020).

Millán et al. (2021) analyzed the critical current density for d-wave superconductors using a generalized Hubbard model on a square lattice, revealing that the d-wave symmetry of the pairing interaction leads to a maximum critical current density (Millán et al., 2021). According to BCS theory, superconductivity arises from the cooperative behavior of conducting electrons, where entangled electron pairs play a crucial role (Bardeen et al., 1957). S.C. Benjamin et al. (2009) highlighted the significance of electron entanglement in solid-state physics for the development of quantum information and computation schemes (Benjamin et al., 2009). Clare Dunning et al. (2005) demonstrated that, in the thermodynamic limit, local concurrences fully describe entanglement in the ground state of the BCS model (Dunning et al., 2005). J. Cordelair et al. (2013) described superconductivity as involving double quantum entanglement, where wave-particles superpose and overlap to form Cooper pairs at the critical temperature (Cordelair et al., 2013). Subir Sachdev (2013) discussed the quantum superposition of electron pairs in Yttrium Barium Copper Oxide (YBCO), noting that removing some oxygen atoms from the YBCO planes increases hole concentration and promotes long-range entanglement of electron pairs (Sachdev, 2013). Gary J. Mooney et al. (2019) used a 20-qubit superconducting quantum computer to demonstrate entanglement in one of the largest solid-state qubit arrays, advancing the implementation of complex quantum algorithms. However, they did not explore the impact of entanglement on the critical magnetic field and critical current density of YBCO (Mooney et al., 2019). Recent research has focused on magnetic fields and critical fields in conventional superconductors, prompting investigations into superconductivity as a quantum entanglement phenomenon and its effects on critical magnetic field and critical current density in unconventional high-Tc YBCO superconductors.

2. Theoretical formulations

2.1. Mathematical Formulations

In GL theory, when the wave function of a particle at a point in space $\left(\psi \right)$ is small and varies slowly in space, the free-energy density (F_s (r)) can be expanded in a power series of the form,

$$F_s(r)=F_n+\alpha {\left|\psi \left(r\right)\right|}^2+{\displaystyle \frac{\beta }{2}}{\left|\psi \left(r\right)\right|}^4+{\displaystyle \frac{1}{{2m}^{*}}}{\left|\left(-i\hslash \nabla -{\displaystyle \frac{e^{*}}{c}}A\left(r\right)\right)\psi (r)\right|}^2+{\displaystyle \frac{{\left|H\right|}^2}{8\pi }}$$

(2.1)

where α and β are phenomenological parameters (β > 0 and the sign of α is temperature dependent), m* = 2m is an effective mass, e* = q* = 2e is the charge of an electron, H is the local magnetic field of the superconductor, A is the magnetic vector potential and B = ∇ X A where B is the external magnetic field. By minimizing the free energy functional in equation (2.1) concerning fluctuations in the order parameter and the vector potential, one arrives at the First GL equation, expressed as, (Kittel, C. (2005, pp 283-286)

$$\alpha \psi +\beta {\left|\psi \right|}^2\psi +{\displaystyle \frac{1}{{2m}^{*}}}{\left(-i\hslash \nabla -{\displaystyle \frac{e^{*}}{c}}A\right)}^2$$

(2.2)

The current density equation, $J_s$, is obtained by minimizing the expression of the first GL Equation concerning vector potential, A, to yield

$$J_s={\displaystyle \frac{1}{2m^{*}}}{\left|-i\hslash \nabla -{\displaystyle \frac{e^{*}}{c}}A\right|}^2{\left|\mathsf{\Psi }\right|}^2$$

(2.3)

In the absence of an external magnetic field, there will be no superconducting current and the equation (2.1) reduces to;

$$F_s-F_n=\alpha \psi +{\displaystyle \frac{\beta }{2}}{\left|\psi \right|}^2\psi$$

(2.4)

This equation is trivial for $\psi =0$ and it corresponds to the normal state where ${T>T}_c$. Below the superconducting transition temperature (Tc), equation (2.5) is expected to have a non-trivial solution (i.e. $\psi \ne 0$) of the form

$${\left|\psi \right|}^2=-{\displaystyle \frac{\alpha }{\beta }}.$$

(2.5a)

For ${T>T}_c$, the expression ${\displaystyle \frac{\alpha \left(T\right)}{\beta }}$ is positive the second part of the equation (2.4) is negative and only $\psi =0$ solves the GL equation. For ${T<T}_c,$ the second part of equation (2.4) is positive and there is no trivial solution for $\psi$. Thus equation (2.5a) can be expressed as

$$\left|\psi \right|={\left({\displaystyle \frac{\alpha (T_c-T)}{\beta }}\right)}^{{\displaystyle \frac{1}{2}}}$$

(2.5b)

Equation (2.5b) yields the GL wave function for a superconducting state, this wave function vanishes at T = Tc, only valid at T > Tc.

The First GL equation (2.2) can be expressed in x-dimensions as

$$\alpha \psi +\beta {\left|\psi \right|}^2\psi +{\displaystyle \frac{1}{{2m}^{*}}}{\left(-i\hslash -{\displaystyle \frac{e^{*}}{c}}A\right)}^2\psi =0$$

(2.6)

In superconductivity, superconducting coherence length is the characteristic exponent of the variations of the density of the superconducting component, (Tinkham, M. 2004, 2^nd Ed.). In the absence of the magnetic field equation (2.3) reduces to,

$$\alpha \psi +\beta {\left|\psi \right|}^3-{\displaystyle \frac{{\hslash }^2}{{2m}^{*}}}{\displaystyle \frac{d^2\psi }{d{x}^2}}=0$$

(2.7)

Assuming $\psi$ is complex and neglecting the term $\beta {\left|\psi \right|}^3$ (not close to the boundary i.e. critical point) in comparison with α, equation (2.7) becomes

$$\alpha \psi ={\displaystyle \frac{{\hslash }^2}{{2m}^{*}}}{\displaystyle \frac{d^2\psi }{d{x}^2}}$$

(2.8)

For the plane wave function, the solution of Equation (2.8) is in the form of,

$$\psi \left(x\right)=exp\left({\displaystyle \frac{ix}{{\xi }_{GL}}}\right)$$

(2.9)

where ${\xi }_{GL}$ is the GL coherence length expressed as a superconducting state (α is negative) yields

$${\xi }_{GL}^2\left(T\right)=-{\displaystyle \frac{{\hslash }^2}{2m^{*}\left|\alpha \right|}}={\displaystyle \frac{{\hslash }^2}{2{{\alpha }_{o}m}^{*}\left(T_c-T\right)}}={\xi }_{GL}^2\left(0\right)$$

(2.10)

Using (2.10), equation (2.7) can be rewritten as

$$\psi +{\displaystyle \frac{\beta }{\left|\alpha \right|}}{\left|\psi \right|}^3-{\xi }_{GL}^2{\displaystyle \frac{d^2\psi }{{dx}^2}}=0$$

(2.11)

We define an arbitrary function f such the wave function $\psi \left(x\right)$ in equation (2.11) is expressed as

$$\psi \left(x\right)=\sqrt{{\displaystyle \frac{\left|\alpha \right|}{\beta }}} g$$

(2.12)

$${\displaystyle \frac{d}{dx}}\left\{-{\displaystyle \frac{1}{2}}{\xi }^2{\left({\displaystyle \frac{dg}{dx}}\right)}^2-{{\displaystyle \frac{1}{2}}g}^2+{{\displaystyle \frac{1}{4}}g}^4\right\}=0$$

(2.13)

If the derivative of the function in (2.13) is zero, then the expression in the brackets is a constant or zero, then

$${\displaystyle \frac{dg}{dx}}={\displaystyle \frac{1}{\sqrt{2} \xi }}\left(1-g^2\right)$$

(2.14)

which as a solution in terms of a hyperbolic function of a tangent, given by

$$g=\tanh {\displaystyle \frac{\mathit{x}}{\sqrt{2}\mathit{\xi }}}$$

(2.15)

Since $\alpha \left(T\right)={\alpha }_o(T-T_c)$, and $\beta >0$, then $\psi \left(x\right)$ defined in equation (2.12) takes the form

$$\psi \left(x\right)={\left({\displaystyle \frac{\left|\alpha \right|}{\beta }}\right)}^{{\displaystyle \frac{1}{2}}}\tanh {\displaystyle \frac{x}{\sqrt{2} \xi }}={\left({\displaystyle \frac{{\mathit{\alpha }}_{\mathit{o}}({\mathit{T}}_{\mathit{c}}-\mathit{T}),}{\mathit{\beta }}}\right)}^{{\displaystyle \frac{1}{2}}}\tanh {\displaystyle \frac{\mathit{x}}{\sqrt{2}\mathit{\xi }}}$$

(2.16)

Equation (2.16) gives an inhomogeneous wave function which is dependent on temperature and space, x. At T = Tc, $\psi =0$ the material reverts to normal state. Also, at x = 0 then $\psi =0$ which makes a superconducting material revert to normal. (Tinkham, M. (2004, 2^nd Ed.)

2.2. Number Density

The order parameter for the entangled superconducting system for two particles that form a cooper pair at a position x₁ and x₂ in space is written in a symmetric form as; (Rashid, M. A. 2019, September 26)

$$\mathsf{\Psi }\left(x_1, x_2\right)={\displaystyle \frac{1}{\sqrt{2}}}\left[{\psi }_{1 }\left(x_1\right){\psi }_2\left(x_2\right)+{\psi }_2\left(x_2\right){ \psi }_1\left(x_1\right)\right]$$

(2.17)

where ${\psi }_1\left(x_1\right)$ and ${\psi }_2\left(x_2\right)$ are the wave functions of the particles in position x₁ and x₂ respectively. Using equation (2.16), $\mathsf{\Psi }(x_1, x_2)$ in (2.17) can be expressed as

$$\mathsf{\Psi }\left(x_1,x_2\right)=\sqrt{{\displaystyle \frac{2\alpha \left(T_c-T\right)}{\beta }}}\left({\displaystyle \frac{\tanh \left(\raisebox{1ex}{$x_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\xi $}\right.\right)+tanh\left(\raisebox{1ex}{$x_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\xi $}\right.\right) }{tanh \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\xi $}\right.\left(x_1+x_2\right)}}-1\right)$$

(2.18)

After applying Hyperbolic functions of sums. Equation (2.18) is the order parameter for an entangled superconducting system of YBCO superconductor. The number density ${\mathit{n}}_s$ of the superconducting electrons, becomes,

$${\mathit{n}}_s={\left|\mathsf{\Psi }({\mathrm{x}}_1,{\mathrm{x}}_2)\right|}^2={\displaystyle \frac{{2a_0}^2{\left(T_c-T\right)}^2}{{\beta }^2}}\left({\displaystyle \frac{{\tanh }^2\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+{tanh}^2\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+2\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{2 }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}}-2{\displaystyle \frac{\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{ }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}}+1\right)$$

(2.18)

3. Ginzburg Landau Penetration Depth

Ginzburg Landau penetration depth is the characteristic length of the fall of the applied magnetic field due to surface currents. It characterizes the typical distance through which the weak magnetic field penetrates a superconductor. (Tinkham, M. (2004) Neglecting the terms $\nabla \mathsf{\Psi }$ and $\nabla {\mathsf{\Psi }}^{*}$ that are not field-related in the current density equation (2.3), to get

$$J_s=-{\displaystyle \frac{e^{*2}}{m^{*}c}}{\left|\mathsf{\Psi }\right|}^{2}A$$

(2.19)

From Maxwell’s equation

$$\nabla \mathrm{X}\mathrm{B}={\displaystyle \frac{4\pi }{c}}{J}_s$$

(2.20)

Taking curl both sides of equation (2.20) and simplify using vector identities, we obtain

$${\nabla }^{2}B=-{\displaystyle \frac{4\pi }{c}}({\nabla X J}_s)={\displaystyle \frac{B}{{\lambda }_{GL\left(T\right)}^2}}$$

(2.21)

where,

$${\lambda }_{GL\left(T\right)}=\sqrt{{\displaystyle \frac{m{c}^2}{8\pi e^2{n}_s}}}$$

(2.22)

is the GL penetration depth, $m^{*}=2m$ and $e^{*}=2e$. Substituting the number density ${\mathit{n}}_s$ equation (2.18) into (2.22), we obtain

$${\lambda }_{GL\left(T\right)}={\lambda }_{GL\left(0\right)}{\left[1-{\displaystyle \frac{T}{T_c}}\right]}^{-{\displaystyle \frac{1}{2}}}$$

(2.23)

where ${\lambda }_{GL\left(0\right)}$ is the penetration depth at zero temperature, T = 0.

$${\lambda }_{GL\left(0\right)}=\sqrt{{\displaystyle \frac{m{c}^2{\beta }^2}{16\pi e^2{a_0}^2{\left(T_c\right)}^2\left(\frac{{\tanh }^2\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+{tanh}^2\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+2\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{2 }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}-2\frac{\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{ }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}+1\right)}}}$$

(2.24)

It follows that the GL penetration depth, equation (2.23) varies as a function of temperature, while the penetration depth at zero temperature, equation (2.24) varies as a function of the distance between the entangled electron pair.

3.1. Critical Magnetic Field

3.1.1. Upper Critical Magnetic Field, ${\mathrm{H}}_{\mathrm{c}2}$

The critical magnetic field is the field that suppresses superconductivity in a material when applied to a superconductor, Kittel, C. (2005, pp 283-286). Since at the transition point of superconductivity, $\psi$ is small then, the first GL Equations (2.6) is linearized to obtain

$${\displaystyle \frac{1}{{2m}^{*}}}{\left(-i\hslash -{\displaystyle \frac{e^{*}}{c}}\mathit{A}\right)}^2\psi =-\alpha \psi$$

(2.25)

At the onset of superconductivity, the magnetic field in the superconducting region is just an applied field, i.e. $A=B\left(0,x,0\right)=B_{x}x$. Hence equation (2.25) becomes

$${\displaystyle \frac{1}{{2m}^{*}}}{\left(-i\hslash -{\displaystyle \frac{e^{*}}{c}}{B}_{x}x\right)}^2\psi =-\alpha \psi$$

(2.26)

Expanding equation (2.26) results in

$$-{\displaystyle \frac{{\hslash }^2}{{2m}^{*}}}\left({\displaystyle \frac{{\partial }^2}{{\partial }_x^2}}\right)\mathsf{\Psi }+{\displaystyle \frac{m^{*}}{2}}{\left({\displaystyle \frac{e^{*}}{m^{*}c}}\mathit{B}x\right)}^2\psi =-\alpha \psi$$

(2.27)

Equation (2.27) is similar to the time-independent Schrodinger equation for a particle of mass $m$ oscillating at a frequency $\omega$, subject to a restoring force $F\left(x\right)=-{m\omega }^{2}x$ is expressed as

$$-{\displaystyle \frac{{\hslash }^2}{{2m}^{*}}}\left({\displaystyle \frac{{\partial }^2}{{\partial }_x^2}}\right)\mathsf{\Psi }+{\displaystyle \frac{{m\omega }^2}{2}}{x}^2\psi =E_n\psi$$

(2.28)

where $E_n$ is the Energy of the Oscillator of frequency expressed as

$$E_n=\left(n+{\displaystyle \frac{1}{2}}\right)\hslash \omega$$

(2.29a)

Comparing the form equation (2.27) and (2.28), we can define the s the cyclotron frequency ${\omega }_c,$ given by

$$\mathsf{\omega }={\omega }_c={\displaystyle \frac{e^{*}B}{{2m}^{*}c}}$$

(2.29b)

The largest value of the magnetic field (B = $B_{max}$) in equation (2.27) corresponds to the smallest eigenvalue n = 0 (ground state energy) in equation (2.28), leading to which the solution of equation (2.28) of the lowest eigenvalue is given by

$$E_0={\displaystyle \frac{1}{2}}\hslash {\omega }_c={\displaystyle \frac{1}{2}}\hslash \left({\displaystyle \frac{e^{*}{B}_{max}}{{2m}^{*}c}}\right)=-\alpha$$

(2.29c)

which simplify to

$$B_{max}=-{\displaystyle \frac{2m^{*}c\left|\alpha \right|}{\hslash e^{*}}}$$

(2.29c)

The highest field in which superconductivity can nucleate in the interior of a bulk sample occurs within the upper critical magnetic field ($H_{c2})$ i.e. $B_{max}=$ $H_{c2}$, then equation (2.29c) can be rewritten as

$$H_{c2}=-{\displaystyle \frac{2m^{*}c\left|\alpha \right|}{\hslash e^{*}}}$$

(2.30)

Substitute for $\alpha$ obtained equation (2.10) into equation (2.30) and simplify, to find the expression for the temperature dependence of the upper critical magnetic field as

$$H_{c2}={\displaystyle \frac{\hslash c}{e^{*}{\xi }_{GL\left(0\right)}^2}}\left(1-{\displaystyle \frac{T}{T_c}}\right)$$

(2.31a)

Taking $T=0$ in equation (2.31a), the expression for the upper critical magnetic field at zero temperature, becomes

$$H_{c2}(0)={\displaystyle \frac{\hslash c{T}_c}{e^{*}{\xi }_{GL\left(0\right)}^2}}$$

(2.31b)

3.1.2. Lower Critical magnetic field, ${\mathrm{H}}_{\mathrm{c}1}$

To obtain the lower critical field, the relation for the penetration depth is used instead of coherence length to obtain the parameter $\alpha$ in equation (2.29d). Using equation (2.5a) to define the number density ${\mathit{n}}_s$, substitute in the penetration depth equation (2.22) and simplify, we find

$$\left|\alpha \right|={\displaystyle \frac{m{c}^2\beta }{8\pi e^2{\lambda }_{GL\left(T\right)}^2}}={\displaystyle \frac{m{c}^2\beta }{8\pi e^2{\lambda }_{GL\left(0\right)}^2}}{\left(1-{\displaystyle \frac{T}{T_c}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(2.32)

after using equation (2.23). Substituting in (2.32) in equation (2.29d), we obtain the lower critical field as

$$H_{c1}={\displaystyle \frac{m^3{c}^2\beta }{4\pi \hslash e^3{\lambda }_{GL\left(0\right)}^2}}{\left(1-{\displaystyle \frac{T}{T_c}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(2.33a)

where ${\lambda }_{GL\left(0\right)}$ is the London penetration depth at zero temperature, equation (2.24). Taking $T=0$ in equation (2.33a), the expression for a lower critical magnetic field at zero temperature becomes

$$H_{c1}(0)={\displaystyle \frac{m^3{c}^2\beta }{4\pi \hslash e^3{\lambda }_{GL\left(0\right)}^2}}$$

(2.33b)

Substituting equation 2.24 in equation 2.33a yields an expression for the dependence of lower critical magnetic field to entanglement distance

$$H_{c1}={\displaystyle \frac{c^2\beta }{2\hslash {\lambda }_{GL\left(T\right)}{e}^2}}{\left({\displaystyle \frac{m^3}{\pi \left(\frac{{\tanh }^2\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+{tanh}^2\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+2\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{2 }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}-2\frac{\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{ }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}+1\right)}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(2.33b)

But ${\lambda }_{GL\left(T\right)}={\lambda }_{GL\left(o\right)}\left({\left(1-{\displaystyle \frac{T}{T_c}}\right)}^{{\displaystyle \frac{1}{2}}}\right)$, hence

$$H_{c1}={\displaystyle \frac{c^2\beta }{2{\hslash \lambda }_{GL\left(0\right)}{e}^2}}{\left({\displaystyle \frac{m^3}{\pi \left(\frac{{\tanh }^2\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+{tanh}^2\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+2\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{2 }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}-2\frac{\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^{ }\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}+1\right)}}\right)}^{{\displaystyle \frac{1}{2}}}{\left(1-{\displaystyle \frac{T}{T_c}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(2.33c)

Equation (2.35c) gives the expression for the temperature dependence of the lower critical magnetic field and the dependence of the lower critical magnetic field with entanglement distance

3.2. Critical Current Density, $J_c$

Critical current density is the value of transport current density which causes intergranular magnetization to collapse at a given magnetic field and temperature, de Gennes, P. G. (1999). Substituting the number density ${\mathit{n}}_s$ equation (2.18) into the current density equation (2.19), to get

$$J_s=-{\displaystyle \frac{{2A{e^{2}a}_0}^2{\left(T_c-T\right)}^2}{mc{\beta }^2}}\left({\displaystyle \frac{{\tanh }^2\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+{tanh}^2\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^2\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}}-2{\displaystyle \frac{\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}}+1\right)$$

(2.34)

At the maximum magnetic field, (at the critical point), the applied field is equal to the vector potential. $H_{c2}=A$, and $J_s=J_c$, hence the critical current density from equation (2.25), becomes

$$J_c=-{\displaystyle \frac{{2H_{c2}{e^{2}a}_0}^2{\left(T_c-T\right)}^2}{mc{\beta }^2}}\left({\displaystyle \frac{{\tanh }^2\left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+{tanh}^2\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{{tanh}^2\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}}-2{\displaystyle \frac{\tanh \left(\raisebox{1ex}{${\mathrm{x}}_1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)+\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\left(\raisebox{1ex}{${\mathrm{x}}_2$}\!\left/ \!\raisebox{-1ex}{$\sqrt{2}\mathsf{\xi }$}\right.\right)}{\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}\frac{1}{\sqrt{2}\xi }\left(x_1+x_2\right)}}+1\right)$$

(2.35)

At zero temperature T = 0, the critical current density becomes,

$$J_c=-{\displaystyle \frac{H_{c2}}{4\pi {\lambda }_{GL\left(0\right)}^2}}{\left(1-{\displaystyle \frac{T^2}{{T_c}^2}}\right)}^{{\displaystyle \frac{1}{2}}}$$

(2.36a)

where $H_{c2}$ is the Upper critical magnetic field and ${\lambda }_{GL\left(0\right)}$ is the London penetration depth at zero temperature, equation (2.24). Taking $T=0$ in equation (2.36a), the expression for critical current density at zero temperature becomes

$$J_c(0)=-{\displaystyle \frac{H_{c2}}{4\pi {\lambda }_{GL\left(0\right)}^2}}$$

(2.36b)

3.2.1. Penetration Depth and Entanglement Distance

Equation (2.23) is used to plot the graph of the penetration depth against temperature at a fixed distance between the entangled electron pair, as shown in Figure 3.3.

Figure 3.3.  

Figure 3.3.  

From Figure 3.3, we observe that there is an increase in the GL penetration depth with temperature (T), and generally, we observe that the penetration depth rises asymptotically as the temperature approaches the critical point. Thus, the penetration of the field increases as the temperature approaches Tc. At Tc, the magnetic field penetrates the YBCO superconductor fully, there is no expulsion of the magnetic field beyond Tc. The penetration of the applied magnetic field increases as the temperature increases, and as the temperature approaches T = Tc, the graph diverges (diamagnetic properties of YBCO suppressed).

3.3. Critical Magnetic Field

a)

Lower Critical Field with Temperature

Equation (2.33a) is used to plot the graph of the lower critical magnetic field for superconducting YBCO against temperature at a fixed distance of $\left(1.8 nanometers\right)$ between the entangled electron pair, as shown in Figure 3.4. We observe in Figure 3.4. that the lower critical magnetic field is non-linearly dependent on the temperature of the entangled YBCO superconductor.

Figure 3.4.  

Figure 3.4.  

As we increase the temperature, the lower critical magnetic field decreases. This happens because the material's capacity to retain superconductivity is dependent on temperature. As temperature decreases, the critical field increases generally to a maximum at absolute zero. At T = Tc, the magnetic field is zero, at this temperature, YBCO has assumed the normal properties of a material.

b)

Lower Critical Field with Entanglement Distance

Equation (2.33b) is used to plot the lower critical magnetic field for superconducting YBCO at zero temperature against the distance between the entangled electron pair, as shown in Figure 3.5.

Figure 3.5.  

Figure 3.5.  

Figure 3.5 shows that as the distance of separation between the entangled pair of electrons increases, the magnetic field also decreases up to about 6.5 nm and then attains a constant value at infinite distances. From the graph, beyond 6.5 nm, the material reverts to normal as there is complete penetration of the magnetic field. We observed that as the temperature of the YBCO superconductor increases, the distance of separation between the copper pairs increases and vice versa.

c)

Upper Critical Field with Temperature

Equation (2.31a) is used to plot the graph of the upper critical magnetic field for superconducting YBCO against temperature at a fixed distance (1.8 nanometers) between the entangled electron pair, as shown in Figure 3.4.

Figure 3.6.  

Figure 3.6.  

From Figure 3.6, we observe that the upper critical magnetic field is linearly dependent on the temperature of the entangled YBCO superconductor. We observe that the upper critical magnetic field decreases as temperature increases and reaches to zero at the critical temperature of superconducting YBCO.

3.4. Critical Current Density, $J_c$

The critical current density J_c is the value of transport current density which causes the intergranular magnetization to collapse at a given magnetic field and temperature. There is a direct relation between the critical field and the critical current, the maximum electric current density that a given superconducting material can carry, before switching into the normal state.

a)

Critical Current Density versus Temperature

Equation (2.36a) is used to plot the graph of the critical current density for superconducting YBCO against temperature at a fixed distance (1.8 nanometers) between the entangled electron pair, as shown in Figure 3.7.

Figure 3.7.  

Figure 3.7.  

It is observed in Figure 3.7 that as the temperature of the YBCO superconductor increases, the critical current density decreases and comes to zero at T = Tc, i.e. at about 93K. Thermal fluctuations strongly affect superconductivity in YBCO at high temperatures. They destroy the critical state as the temperature approaches Tc.

b)

Critical Current Density versus Temperature

Equation (2.36b) is used to plot the critical current density for superconducting YBCO at zero temperature against the distance between the entangled electron pair, as shown in Figure 3.8.

Figure 3.8.  

Figure 3.8.  

From Figure 3.8 we see that, between x = 1 nm to x = 1.8 nm, the current flows in a material in a normal state. When the distance reaches 1.8, it increases non-linearly with increasing entanglement distance. The critical current increases from x = 1.8 nm to about x = 7.8 nm which flattens for infinite distances having attained a constant value. Therefore, the critical current in the entangled YBCO superconductor increases with the increase of entanglement distance from x = 1.8 nm to x = 7.8, posting a maximum value at x = 7.8.

4.1. Conclusions

The main purpose of this research work is to assess the effect of entanglement on the critical magnetic field and critical current density of superconducting YBCO by using the Ginzburg-Landau approach. From the calculations, the phase diagrams are plotted by using MATLAB scripts and Mathcad prime 9.0.0.0 scripts. From the results, it's clear that the distance of separation between the electrons affects the current density and critical magnetic field of a superconductor. In the case of YBCO, the critical current increases non-linearly when the distance of separation ranges from x = 1.8 nm up to about x = 7.8 nm which flattens for higher and infinite distances attaining a constant value. Also, the penetration depth varies with the distance of separation of the entangled electrons; in this case, it is noted that generally, the penetration depth of a superconductor decreases as the entanglement distance increases attaining a minimum value of 0.168 nanometers at infinite separation distances. Consequently, the critical magnetic field decreases non-linearly when the distance of separation ranges from 1.5 nm up to 6.5 nm where it flattens for higher separation distances.

From the phase diagrams plotted, it can be concluded that the upper critical magnetic field of superconducting YBCO is inversely related to temperature which is in agreement with experimental observations, critical current density, Lower critical magnetic field, and penetration depth are non-linearly dependent on temperature. Also, the phase diagrams from the dependence of entanglement distance to the critical field, critical current density, GL penetration depth, and GL coherence length are assessed. In a nutshell, the entanglement distance affects greatly the critical fields and current densities of an entangled YBCO superconductor.

4.2. Recommendations

This research focused on the effect of entanglement on penetration depth, critical current density, and lower critical magnetic field. The coherence length and upper critical magnetic field were not related to entanglement distance, further studies are recommended to investigate the effect of entanglement on the coherence length and upper critical magnetic field. Our model of entanglement of electron-pair (cooper pair) shows reasonable agreement with available theoretical data in determining the effect of the entanglement on YBCO critical properties. The theory of entanglement in high-temperature superconductivity should be explored as it may unleash remarkable features on the properties of superconductors. The efforts in determining the pairing mechanism as an entanglement in this system need to continue, for such efforts go hand in hand with enhancing prospects for new High-Temperature Superconductor materials and novel applications in power industries, transportation industries, and medicine.