sc-vortex-2d

This library provide numerical results of s-wave Caroli-de Gennes-Matricon mode (CdGM mode) at $T/T_c=0.3, 0.5, 0.8$. You can get the eigenenergy, eigenfunction and pair potential at each temperature.

Description

It is known that there are low-energy excitation levels inside the s-wave vortex core in the superconductor. These states are got by solving following Bogoliubov-de Gennes equation(BdG eq) self-consistently.

$$\left[-\frac{1}{2k_{F}\xi_{0}}\nabla^{2}-\mu\right]\mathcal{U}_{q}(\boldsymbol{r})+\Delta(\boldsymbol{r})\mathcal{V}_{q}(\boldsymbol{r}) = E_{q}\mathcal{U}_{q}(\boldsymbol{r})$$ $$\left[\frac{1}{2k_{F}\xi_{0}}\nabla^{2}+\mu\right]\mathcal{V}_{q}(\boldsymbol{r})+\Delta^{*}(\boldsymbol{r})\mathcal{U}_{q}(\boldsymbol{r}) = E_{q}\mathcal{V}_{q}(\boldsymbol{r})$$ $$\Delta(\boldsymbol{r})=g\sum_{E_{q}\leq E_{\mathrm{c}}} \mathcal{U}_{q}(r)\mathcal{V}_{q}^{*}(r)[1-2f(E_{q})]$$

Here, BdG eq is rewritten in dimensionless form using Pippard length $\xi_{0} = \hbar v_{F}/\Delta_{0}$ and zero-temperature bulk gap $\Delta_{0}$. $f(E_{q})$ is Fermi distribution function. Solutions in an isolated vortex, especially CdGM mode is given by following form.

$$\begin{bmatrix} \mathcal{U}_{n}(r, \theta) \\\ \mathcal{V}_{n}(r, \theta) \end{bmatrix} =\frac{1}{\sqrt{2\pi}} \begin{bmatrix} u_{n}(r)e^{in\theta} \\\ v_{n}(r)e^{i(n + 1)\theta} \end{bmatrix}$$

Here, $n$ corresponds to angular momentum quantum number, i.e. CdGM mode is characterized by this number. In this library, the range of $n$ is integers of in $[-100, 99]$. Note that the part of $u_{n}(r), v_{n}(r)$ in the right side of above formula is one of the target of this library, not the left side of it.

Install

You can install via PyPI. For example,

 $ pip install  sc-vortex-2d 

Usage

"""Sample python code"""

from sc_vortex_2d.vortex import VortexInstanceT03
from scipy.interpolate import CubicSpline

instance: VortexInstanceT03 = VortexInstanceT03()

delta: CubicSpline = instance.get_pair_potential() 

e0: float = instance.get_ith_eigen_energy(0) # lowest energy level in the region of e > 0.

u0, v0 = instance.get_ith_eigen_func(0) # get wave functions

params = instance.Parameters # Parameters of the system. Enum.

Links

• https://webpark1378.sakura.ne.jp/
• F. Gygi and M. Schlüter, Phys. Rev. B, 41, 822 (1990)