spectrum.rst

Beyond transport: Band structure and closed systems

Band structure calculations

When doing transport simulations, one also often needs to know the band structure of the leads, i.e. the energies of the propagating plane waves in the leads as a function of momentum. This band structure contains information about the number of modes, their momenta and velocities.

In this example, we aim to compute the band structure of a simple tight-binding wire.

Computing band structures in Kwant is easy. Just define a lead in the usual way:

"Usual way" means defining a translational symmetry vector, as well as one unit cell of the lead, and the hoppings to neighboring unit cells. This information is enough to make the infinite, translationally invariant system needed for band structure calculations.

In the previous examples ~kwant.builder.Builder instances like the one created above were attached as leads to the Builder instance of the scattering region and the latter was finalized. The thus created system contained implicitly finalized versions of the attached leads. However, now we are working with a single lead and there is no scattering region. Hence, we have to finalize the Builder of our sole lead explicitly.

That finalized lead is then passed to ~kwant.plotter.bands. This function calculates energies of various bands at a range of momenta and plots the calculated energies. It is really a convenience function, and if one needs to do something more profound with the dispersion relation these energies may be calculated directly using ~kwant.physics.Bands. For now we just plot the bandstructure:

This gives the result:

where we observe the cosine-like dispersion of the square lattice. Close to k=0 this agrees well with the quadratic dispersion this tight-binding Hamiltonian is approximating.

Closed systems

Although Kwant is (currently) mainly aimed towards transport problems, it can also easily be used to compute properties of closed systems -- after all, a closed system is nothing more than a scattering region without leads!

In this example, we compute the wave functions of a closed circular quantum dot and its spectrum as a function of magnetic field (Fock-Darwin spectrum).

To compute the eigenenergies and eigenstates, we will make use of the sparse linear algebra functionality of SciPy, which interfaces the ARPACK package:

We set up the system using the shape-function as in :ref:`tutorial-abring`, but do not add any leads:

We add the magnetic field using a function and a global variable as we did in the two previous tutorial. (Here, the gauge is chosen such that A_x(y) = - B y and A_y=0.)

The spectrum can be obtained by diagonalizing the Hamiltonian of the system, which in turn can be obtained from the finalized system using ~kwant.system.System.hamiltonian_submatrix:

Note that we use sparse linear algebra to efficiently calculate only a few lowest eigenvalues. Finally, we obtain the result:

At zero magnetic field several energy levels are degenerate (since our quantum dot is rather symmetric). These degeneracies are split by the magnetic field, and the eigenenergies flow towards the Landau level energies at higher magnetic fields [1].

The eigenvectors are obtained very similarly, and can be plotted directly using ~kwant.plotter.map:

The last two arguments to ~kwant.plotter.map are optional. The first prevents a colorbar from appearing. The second, oversampling=1, makes the image look better for the special case of a square lattice.

As our model breaks time reversal symmetry (because of the applied magnetic field) we can also see an interesting property of the eigenstates, namely that they can have non-zero local current. We can calculate the local current due to a state by using kwant.operator.Current and plotting it using kwant.plotter.current:

Footnotes

[1]	Again, in this tutorial example no care was taken into choosing appropriate material parameters or units. For this reason, magnetic field is given only in "arbitrary units".