Critical temperature in a NSN structure (MPI-parallelized)

Consider a NSN structure:

_images/aafig-2ef8bb4fb7a853fa5e745b22d61e56e09c9689f6.png

where the wire (of length L_s) between the N-terminals is superconducting. Because of the normal terminals, the critical temperature of the superconducting wire is less than the bulk T_c. One can easily show that the critical temperature here is given as a solution of

(1)\frac{1}{\lambda} = \psi\left(\frac{3}{2}+\frac{\alpha}{4\pi T_c} + \frac{\omega_D}{2\pi T_c}\right) - \psi\left(\frac{1}{2} + \frac{\alpha}{4\pi T_c}\right)

where \alpha=\pi^2/L_s^2 is the smallest eigenvalue of -\nabla^2, \psi is the digamma function, \lambda the superconducting coupling constant, and \omega_D the cutoff frequency. Keeping \Delta(T=0)=\Delta_0 constant, (1) approaches

(2)\ln\frac{\Delta_0}{4\pi T_c} = \psi(\frac{1}{2} + \frac{\alpha}{4\pi T_c})

when \omega_D\rightarrow\infty. For \alpha=0, the usual BCS result T_c=\Delta_0 e^\gamma/\pi is retrieved.

We can also compute this temperature numerically, by making use of the self-consistent iteration implemented in functions self_consistent_matsubara_iteration() or self_consistent_realtime_iteration(). The code for this computation is given in example-nsn-tc.py

A multicore version (without MPI) of the same script is in example-nsn-tc-multiprocessing.py

This code also serves as an example on how to write MPI-parallelized code with the mpi4py Python module. usadel1 does not itself support parallel computation, but sweeps over parameter ranges can be distributed to multiple processors. To run this on 8 processors in parallel, you can do:

mpirun -np 8 python example-nsn-tc.py

The mpi4py module exposes a standard MPI interface — if you are familiar with how MPI parallel codes work in general, you should have no problems understanding what is going on. If you want to learn more about MPI, many guides exist — for example “Rinnakkaisohjelmointi MPI:llä” by CSC Finland (in Finnish), or the book “Parallel Programming with MPI” by P.S. Pachero.

The result is shown in the figure below:

_images/nsn-tc.png

The wire is superconducting only for L_s > \pi \xi_0.

See also

example-nsn-tc.py example-nsn-tc-multiprocessing.py