Question Statement
In a Type-II superconductor placed in a magnetic field $B$ within the vortex (mixed) state ($H_{c1} < H < H_{c2}$), the magnetic flux penetrates the material in the form of quantized magnetic flux vortices (fluxons). What is the value of the magnetic flux quantum $\Phi_0$?
(A)
$\frac{h}{2e} \approx 2.07 \times 10^{-15}\text{ Wb}$
(B)
$\frac{h}{e} \approx 4.14 \times 10^{-15}\text{ Wb}$
(C)
$\frac{e}{h} \approx 2.42 \times 10^{14}\text{ Wb}^{-1}$
(D)
$\frac{2h}{e} \approx 8.27 \times 10^{-15}\text{ Wb}$
Step-by-Step KaTeX Derivation & Detailed Explanation
### Step-by-Step Derivation & Verification: 1. **Ginzburg-Landau Order Parameter**: $\psi(\vec{r}) = \sqrt{n_s} e^{i\theta(\vec{r})}$. 2. **Supercurrent Density Relation**: $\vec{J}_s = \frac{q^* \hbar}{m^*} n_s \nabla\theta - \frac{(q^*)^2}{m^*} n_s \vec{A}$ 3. **Contour Integration Deep in Bulk** ($\vec{J}_s = 0$): $\hbar \oint \nabla\theta \cdot d\vec{l} = q^* \oint \vec{A} \cdot d\vec{l} = q^* \Phi$ 4. **Single-Valuedness Requirement**: Phase change around a closed loop must be an integer multiple of $2\pi$: $\oint \nabla\theta \cdot d\vec{l} = 2\pi n \implies \hbar (2\pi n) = q^* \Phi \implies \Phi = n \frac{h}{q^*}$ 5. **Cooper Pair Charge**: $q^* = 2e$: $\Phi_0 = \frac{h}{2e} = \frac{6.626 \times 10^{-34}\text{ J}\cdot\text{s}}{2(1.602 \times 10^{-19}\text{ C})} \approx 2.0678 \times 10^{-15}\text{ Wb}$ 6. **Conclusion**: Option (A) is correct.