Kohn-Sham Bogoliubov-de Gennes (KS-BdG) simulations allow to combine the accurate electronic structure based on DFT with a description for superconductivity leading to an inherent multi-band description of superconductivity in real materials.
> Further information on the method and the underlying implementation can be found in the KS-BdG method paper:
> [P. Rüßmannn and S. Blügel, Phys. Rev. B **105**, 125143 (2022)](https://journals.aps.org/prb/abstract/10.1103/PhysRevB.105.125143)
> <details><summary>Recent applications</summary>
>
> - [K. Janßen, P. Rüßmann, et al., Phys. Rev. Materials **8**, 034205 (2024)](https://link.aps.org/doi/10.1103/PhysRevMaterials.8.034205)
*Semi-circle energy contour between `EMIN` and `EFERMI`. The energy mesh gets exponentially denser when approaching the Fermi level. The colour indicates the k-mesh that is limited to 4 different sets of k-points (needed to avoid undersampling of the BZ integration).*
3. We then construct a starting value for the superconducting pairing potential.
This is done in a one-shot calculation that uses a large value for `Delta_BdG` in
the input. The output is an anomalous density (called `den_lm_ir.001.1.dat`) which
is then read and updated in the following steps. This step is done in `2_init_BdG_one_shot`.
4. Then we converge the BdG calculation in `3_BdG_scf`. This then produces a new output
potential and updates the anomalous density `den_lm_ir.*`.
*First line shows where Broyden mixing of the superconducting pairing potential $\Delta$ starts (simple mixing is done until then) and the second line indicates when Broyden mixing of the potential starts. The idea is to first converge $\Delta$ and then let the potential follow.*
5. Finally, in `4_BdG_dos` the DOS in the superconducting state is computed.
*BdG-DOS in the superconducting state. A denser k-mesh (e.g. 400x400x400) would be needed to reduce the numerical noise seen for example between -5 and -10 meV.*
In practice the steps 4. and 5. are iterated with different values of the coupling
constant `LAMBDA_BdG` to find a gap size that matches the experiment.
*Change in the gap size with the coupling constant.*
The notebooks `Convergence_plots.ipynb` and `DOS_plot.ipynb` show the outcome of this example.
### Limitations
So far we can only do s-wave coupling.
With SOC the calculation takes much longer because of the required large k-mesh
and the lower symmetries in the Brillouin zone integration.
Non-collinear magnetism does not work with BdG, make sure to fix the directions of the angles
using the `<FIX_NONCO_ANGLES>= True` input (also if you want to use SOC!).
### Download
- The example can be downloaded from [here](https://iffcloud.fz-juelich.de/s/r9a7z3Z6mR9WyES).
- This example can also be done with AiiDA-KKR and the corresponding tutorial and AiiDA export file can be downloaded [here](https://iffcloud.fz-juelich.de/s/jNtpp4jBa8PDG3e).
