tutorial4.rst

Using a more complicated lattice (graphene)

In the following example, we are going to calculate the conductance through a graphene quantum dot with a p-n junction and two non-collinear leads. In the process, we will touch all of the topics that we have seen in the previous tutorials, but now for the honeycomb lattice. As you will see, everything carries over nicely.

We begin by defining the honeycomb lattice of graphene. This is in principle already done in kwant.lattice.Honeycomb, but we do it explicitly here to show how to define a new lattice:

The first argument to the ~kwant.lattice.make_lattice function is the list of primitive vectors of the lattice; the second one is the coordinates of basis atoms. The honeycomb lattice has two basis atoms. Each type of basis atom by itself forms a regular lattice of the same type as well, and those sublattices are referenced as a and b above.

In the next step we define the shape of the scattering region (circle again) and add all lattice points using the shape()-functionality:

As you can see, this works exactly the same for any kind of lattice. We add the onsite energies using a function describing the p-n junction; in contrast to the previous tutorial, the potential value is this time taken from the scope of make_system(), since we keep the potential fixed in this example.

As a next step we add the hoppings, making use of ~kwant.builder.Builder.possible_hoppings. Since we use our home-made lattice (instead of kwant.lattice.Honeycomb), we have to define the hoppings ourselves:

The nearest-neighbor model for graphene contains only hoppings between different basis atoms. For these type of hoppings, it is not enough to specify relative lattice indices, but we also need to specify the proper target and source sublattices. Remember that the format of the hopping specification is (i,j), target, source. In the previous examples (i.e. :ref:`tutorial_spinorbit`) target=source=lat, whereas here we have to specify different sublattices. Furthermore, note that the directions given by the lattice indices (1, 0) and (0, 1) are not orthogonal any more, as they are given with respect to the two primitive vectors [(1, 0), (sin_30, cos_30)].

Adding the hoppings however still works the same way:

Modifying the scattering region is also possible as before. Let's do something crazy, and remove an atom in sublattice A (which removes also the hoppings from/to this site) as well as add an additional link:

Note again that the conversion from a tuple (i,j) to site is done by the sublattices a and b.

The leads are defined almost as before:

Note the method ~kwant.lattice.PolyatomicLattice.vec used in calculating the parameter for ~kwant.lattice.TranslationalSymmetry. The latter expects a real-space symmetry vector, but for many lattices symmetry vectors are more easily expressed in the natural coordinate system of the lattice. The vec method of lattices maps a lattice vector to a real-space vector.

Observe also that the translational vectors graphene.vec((-1, 0)) and graphene.vec((0, 1)) are not orthogonal any more as they would have been in a square lattice -- they follow the non-orthogonal primitive vectors defined in the beginning.

Later, we will compute some eigenvalues of the closed scattering region without leads. This is why we postpone attaching the leads to the system. Instead, we return the scattering region and the leads separately.

The computation of some eigenvalues of the closed system is done in the following piece of code:

Here we use in contrast to the previous example a sparse matrix and the sparse linear algebra functionality of scipy (this requires scipy version >= 0.9.0; since the remaining part of the example does not depend on this eigenenergy calculation, a try-block simply skips this calculation if a lower scipy version is installed.)

The code for computing the band structure and the conductance is identical to the previous examples, and needs not be further explained here.

Finally, in the main() function we make and plot the system:

We customize the plotting: we set the site_colors argument of ~kwant.plotter.plot to a function which returns 0 for sublattice a and 1 for sublattice b:

def group_colors(site):
    return 0 if site.group == a else 1

The function ~kwant.plotter.plot shows these values using a color scale (grayscale by default). The symbol size is specified in points, and is independent on the overall figure size.

Plotting the closed system gives this result:

Computing the eigenvalues of largest magnitude,

should yield two eigenvalues similar to [ 3.07869311 +1.02714523e-17j, -3.06233144 -6.68085759e-18j] (round-off might change the imaginary part which would be equal to zero for exact arithmetics).

The remaining code of main() attaches the leads to the system and plots it again:

It computes the band structure of one of lead 0:

showing all the features of a zigzag lead, including the flat edge state bands (note that the band structure is not symmetric around zero energy, as we have a potential in the leads).

Finally the transmission through the system is computed,

showing the typical resonance-like transmission probability through an open quantum dot