Plotting Kwant systems and data in various styles
-------------------------------------------------
The plotting functionality of Kwant has been used extensively (through
`~kwant.plotter.plot` and `~kwant.plotter.map`) in the previous tutorials. In
addition to this basic use, `~kwant.plotter.plot` offers many options to change
the plotting style extensively. It is the goal of this tutorial to show how
these options can be used to achieve various very different objectives.
2D example: graphene quantum dot
................................
.. seealso::
You can execute the code examples live in your browser by
activating thebelab:
.. thebe-button:: Activate Thebelab
.. seealso::
The complete source code of this example can be found in
:jupyter-download:script:`plot_graphene`
.. jupyter-kernel::
:id: plot_graphene
.. jupyter-execute::
:hide-code:
# Tutorial 2.8.1. 2D example: graphene quantum dot
# ================================================
#
# Physics background
# ------------------
# - graphene edge states
#
# Kwant features highlighted
# --------------------------
# - demonstrate different ways of plotting
import warnings
warnings.simplefilter("ignore")
from matplotlib import pyplot
import kwant
.. jupyter-execute:: boilerplate.py
:hide-code:
We begin by first considering a circular graphene quantum dot (similar to what
has been used in parts of the tutorial :ref:`tutorial-graphene`.) In contrast
to previous examples, we will also use hoppings beyond next-nearest neighbors:
.. jupyter-execute::
lat = kwant.lattice.honeycomb(norbs=1)
a, b = lat.sublattices
def make_system(r=8, t=-1, tp=-0.1):
def circle(pos):
x, y = pos
return x**2 + y**2 < r**2
syst = kwant.Builder()
syst[lat.shape(circle, (0, 0))] = 0
syst[lat.neighbors()] = t
syst.eradicate_dangling()
if tp:
syst[lat.neighbors(2)] = tp
return syst
Note that adding hoppings hoppings to the `n`-th nearest neighbors can be
simply done by passing `n` as an argument to
`~kwant.lattice.Polyatomic.neighbors`. Also note that we use the method
`~kwant.builder.Builder.eradicate_dangling` to get rid of single atoms sticking
out of the shape. It is necessary to do so *before* adding the
next-nearest-neighbor hopping [#]_.
Of course, the system can be plotted simply with default settings,
however, due to the richer structure of the lattice, this results in a rather
busy plot:
.. jupyter-execute::
syst = make_system()
kwant.plot(syst);
A much clearer plot can be obtained by using different colors for both
sublattices, and by having different line widths for different hoppings. This
can be achieved by passing a function to the arguments of
`~kwant.plotter.plot`, instead of a constant. For properties of sites, this
must be a function taking one site as argument, for hoppings a function taking
the start end end site of hopping as arguments:
.. jupyter-execute::
def family_color(site):
return 'black' if site.family == a else 'white'
def hopping_lw(site1, site2):
return 0.04 if site1.family == site2.family else 0.1
kwant.plot(syst, site_lw=0.1, site_color=family_color, hop_lw=hopping_lw);
Note that since we are using an unfinalized Builder, a ``site`` is really an
instance of `~kwant.system.Site`. With these adjustments we arrive at a plot
that carries the same information, but is much easier to interpret.
Apart from plotting the *system* itself, `~kwant.plotter.plot` can also be used
to plot *data* living on the system.
As an example, we now compute the eigenstates of the graphene quantum dot and
intend to plot the wave function probability in the quantum dot. For aesthetic
reasons (the wave functions look a bit nicer), we restrict ourselves to
nearest-neighbor hopping. Computing the wave functions is done in the usual
way (note that for a large-scale system, one would probably want to use sparse
linear algebra):
.. jupyter-execute::
import scipy.linalg as la
syst = make_system(tp=0).finalized()
ham = syst.hamiltonian_submatrix()
evecs = la.eigh(ham)[1]
wf = abs(evecs[:, 225])**2
In most cases, to plot the wave function probability, one wouldn't use
`~kwant.plotter.plot`, but rather `~kwant.plotter.map`. Here, we plot the
`n`-th wave function using it.
This results in a standard pseudocolor plot, showing in this case (``n=225``) a
graphene edge state, i.e. a wave function mostly localized at the zigzag edges
of the quantum dot.
.. jupyter-execute::
kwant.plotter.map(syst, wf, oversampling=10, cmap='gist_heat_r');
However although in general preferable, `~kwant.plotter.map` has a few
deficiencies for this small system: For example, there are a few distortions at
the edge of the dot. (This cannot be avoided in the type of interpolation used
in `~kwant.plotter.map`). However, we can also use `~kwant.plotter.plot` to
achieve a similar, but smoother result.
For this note that `~kwant.plotter.plot` can also take an array of floats (or
function returning floats) as value for the ``site_color`` argument (the same
holds for the hoppings). Via the colormap specified in ``cmap`` these are mapped
to color, just as `~kwant.plotter.map` does! In addition, we can also change
the symbol shape depending on the sublattice. With a triangle pointing up and
down on the respective sublattice, the symbols used by plot fill the space
completely:
.. jupyter-execute::
def family_shape(i):
site = syst.sites[i]
return ('p', 3, 180) if site.family == a else ('p', 3, 0)
def family_color(i):
return 'black' if syst.sites[i].family == a else 'white'
kwant.plot(syst, site_color=wf, site_symbol=family_shape,
site_size=0.5, hop_lw=0, cmap='gist_heat_r');
Note that with ``hop_lw=0`` we deactivate plotting the hoppings (that would not
serve any purpose here). Moreover, ``site_size=0.5`` guarantees that the two
different types of triangles touch precisely: By default, `~kwant.plotter.plot`
takes all sizes in units of the nearest-neighbor spacing. ``site_size=0.5``
thus means half the distance between neighboring sites (and for the triangles
this is interpreted as the radius of the inner circle).
Finally, note that since we are dealing with a finalized system now, a site `i`
is represented by an integer. In order to obtain the original
`~kwant.system.Site`, ``syst.sites[i]`` can be used.
The way how data is presented of course influences what features of the data
are best visible in a given plot. With `~kwant.plotter.plot` one can easily go
beyond pseudocolor-like plots. For example, we can represent the wave function
probability using the symbols itself:
.. jupyter-execute::
def site_size(i):
return 3 * wf[i] / wf.max()
kwant.plot(syst, site_size=site_size, site_color=(0, 0, 1, 0.3),
hop_lw=0.1);
Here, we choose the symbol size proportional to the wave function probability,
while the site color is transparent to also allow for overlapping symbols to be
visible. The hoppings are also plotted in order to show the underlying lattice.
.. rubric:: Footnotes
.. [#] A dangling site is defined as having only one hopping connecting it to
the rest. With next-nearest-neighbor hopping also all sites that are
dangling with only nearest-neighbor hopping have more than one hopping.
3D example: zincblende structure
................................
.. seealso::
The complete source code of this example can be found in
:jupyter-download:script:`plot_zincblende`
.. jupyter-kernel::
:id: plot_zincblende
.. jupyter-execute::
:hide-code:
# Tutorial 2.8.2. 3D example: zincblende structure
# ================================================
#
# Physical background
# -------------------
# - 3D Bravais lattices
#
# Kwant features highlighted
# --------------------------
# - demonstrate different ways of plotting in 3D
from matplotlib import pyplot
import kwant
.. jupyter-execute:: boilerplate.py
:hide-code:
Zincblende is a very common crystal structure of semiconductors. It is a
face-centered cubic crystal with two inequivalent atoms in the unit cell
(i.e. two different types of atoms, unlike diamond which has the same crystal
structure, but two equivalent atoms per unit cell).
It is very easily generated in Kwant with `kwant.lattice.general`:
.. jupyter-execute::
lat = kwant.lattice.general([(0, 0.5, 0.5), (0.5, 0, 0.5), (0.5, 0.5, 0)],
[(0, 0, 0), (0.25, 0.25, 0.25)],
norbs=1)
a, b = lat.sublattices
Note how we keep references to the two different sublattices for later use.
A three-dimensional structure is created as easily as in two dimensions, by
using the `~kwant.lattice.PolyatomicLattice.shape`-functionality:
.. jupyter-execute::
def make_cuboid(a=15, b=10, c=5):
def cuboid_shape(pos):
x, y, z = pos
return 0 <= x < a and 0 <= y < b and 0 <= z < c
syst = kwant.Builder()
syst[lat.shape(cuboid_shape, (0, 0, 0))] = None
syst[lat.neighbors()] = None
return syst
We restrict ourselves here to a simple cuboid, and do not bother to add real
values for onsite and hopping energies, but only the placeholder ``None`` (in a
real calculation, several atomic orbitals would have to be considered).
`~kwant.plotter.plot` can plot 3D systems just as easily as its two-dimensional
counterparts:
.. jupyter-execute::
syst = make_cuboid()
kwant.plot(syst);
You might notice that the standard options for plotting are quite different in
3D than in 2D. For example, by default hoppings are not printed, but sites are
instead represented by little "balls" touching each other (which is achieved by
a default ``site_size=0.5``). In fact, this style of plotting 3D shows quite
decently the overall geometry of the system.
When plotting into a window, the 3D plots can also be rotated and scaled
arbitrarily, allowing for a good inspection of the geometry from all sides.
.. note::
Interactive 3D plots usually do not have the proper aspect ratio, but are a
bit squashed. This is due to bugs in matplotlib's 3D plotting module that
does not properly honor the corresponding arguments. By resizing the plot
window however one can manually adjust the aspect ratio.
Also for 3D it is possible to customize the plot. For example, we
can explicitly plot the hoppings as lines, and color sites differently
depending on the sublattice:
.. jupyter-execute::
syst = make_cuboid(a=1.5, b=1.5, c=1.5)
def family_colors(site):
return 'r' if site.family == a else 'g'
kwant.plot(syst, site_size=0.18, site_lw=0.01, hop_lw=0.05,
site_color=family_colors);
which results in a 3D plot that allows to interactively (when plotted
in a window) explore the crystal structure.
Hence, a few lines of code using Kwant allow to explore all the different
crystal lattices out there!
.. note::
- The 3D plots are in fact only *fake* 3D. For example, sites will always
be plotted above hoppings (this is due to the limitations of matplotlib's
3d module)
- Plotting hoppings in 3D is inherently much slower than plotting sites.
Hence, this is not done by default.