# 2.5. Beyond square lattices: graphene#

See also

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See also

The complete source code of this example can be found in
`graphene.py`

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:

```
# Tutorial 2.5. Beyond square lattices: graphene
# ==============================================
#
# Physics background
# ------------------
# Transport through a graphene quantum dot with a pn-junction
#
# Kwant features highlighted
# --------------------------
# - Application of all the aspects of tutorials 1-3 to a more complicated
# lattice, namely graphene
from math import pi, sqrt, tanh
from matplotlib import pyplot
import kwant
# For computing eigenvalues
import scipy.sparse.linalg as sla
sin_30, cos_30 = (1 / 2, sqrt(3) / 2)
```

```
import matplotlib
import matplotlib.pyplot
from matplotlib_inline.backend_inline import set_matplotlib_formats
matplotlib.rcParams['figure.figsize'] = matplotlib.pyplot.figaspect(1) * 2
set_matplotlib_formats('svg')
```

```
graphene = kwant.lattice.general([(1, 0), (sin_30, cos_30)],
[(0, 0), (0, 1 / sqrt(3))],
norbs=1)
a, b = graphene.sublattices
```

The first argument to the `general`

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:

```
r = 10
w = 2.0
pot = 0.1
```

```
#### Define the scattering region. ####
# circular scattering region
def circle(pos):
x, y = pos
return x ** 2 + y ** 2 < r ** 2
syst = kwant.Builder()
# w: width and pot: potential maximum of the p-n junction
def potential(site):
(x, y) = site.pos
d = y * cos_30 + x * sin_30
return pot * tanh(d / w)
syst[graphene.shape(circle, (0, 0))] = potential
```

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
`HoppingKind`

. For illustration purposes we define
the hoppings ourselves instead of using `graphene.neighbors()`

:

```
# specify the hoppings of the graphene lattice in the
# format expected by builder.HoppingKind
hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b))
```

The nearest-neighbor model for graphene contains only
hoppings between different basis atoms. For this 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.
Matrix structure of on-site and hopping elements) `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 anymore, since 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:

```
syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
```

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:

```
# Modify the scattering region
del syst[a(0, 0)]
syst[a(-2, 1), b(2, 2)] = -1
```

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:

```
#### Define the leads. ####
# left lead
sym0 = kwant.TranslationalSymmetry(graphene.vec((-1, 0)))
def lead0_shape(pos):
x, y = pos
return (-0.4 * r < y < 0.4 * r)
lead0 = kwant.Builder(sym0)
lead0[graphene.shape(lead0_shape, (0, 0))] = -pot
lead0[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
# The second lead, going to the top right
sym1 = kwant.TranslationalSymmetry(graphene.vec((0, 1)))
def lead1_shape(pos):
v = pos[1] * sin_30 - pos[0] * cos_30
return (-0.4 * r < v < 0.4 * r)
lead1 = kwant.Builder(sym1)
lead1[graphene.shape(lead1_shape, (0, 0))] = pot
lead1[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
```

Note the method `vec`

used in calculating the
parameter for `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 is thus used to map 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.

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

```
def compute_evs(syst):
# Compute some eigenvalues of the closed system
sparse_mat = syst.hamiltonian_submatrix(sparse=True)
evs = sla.eigs(sparse_mat, 2)[0]
print(evs.real)
```

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

Finally, we plot the system:

```
def plot_conductance(syst, energies):
# Compute transmission as a function of energy
data = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
data.append(smatrix.transmission(0, 1))
pyplot.figure()
pyplot.plot(energies, data)
pyplot.xlabel("energy [t]")
pyplot.ylabel("conductance [e^2/h]")
pyplot.show()
def plot_bandstructure(flead, momenta):
bands = kwant.physics.Bands(flead)
energies = [bands(k) for k in momenta]
pyplot.figure()
pyplot.plot(momenta, energies)
pyplot.xlabel("momentum [(lattice constant)^-1]")
pyplot.ylabel("energy [t]")
pyplot.show()
```

```
# To highlight the two sublattices of graphene, we plot one with
# a filled, and the other one with an open circle:
def family_colors(site):
return 0 if site.family == a else 1
# Plot the closed system without leads.
kwant.plot(syst, site_color=family_colors, site_lw=0.1, colorbar=False);
```

We customize the plotting: we set the *site_colors* argument of
`plot`

to a function which returns 0 for
sublattice *a* and 1 for sublattice *b*:

```
def family_colors(site):
return 0 if site.family == a else 1
```

The function `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.

Computing the eigenvalues of largest magnitude,

```
compute_evs(syst.finalized())
```

```
[ 3.07869311 -3.06233144]
```

yields two eigenvalues equal to ```
[ 3.07869311,
-3.06233144]
```

.

The remaining code attaches the leads to the system and plots it again:

```
# Attach the leads to the system.
for lead in [lead0, lead1]:
syst.attach_lead(lead)
# Then, plot the system with leads.
kwant.plot(syst, site_color=family_colors, site_lw=0.1,
lead_site_lw=0, colorbar=False);
```

We then finalize the system:

```
syst = syst.finalized()
```

and compute the band structure of one of lead 0:

```
# Compute the band structure of lead 0.
momenta = [-pi + 0.02 * pi * i for i in range(101)]
plot_bandstructure(syst.leads[0], momenta)
```

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, due to a potential in the leads).

Finally the transmission through the system is computed,

```
# Plot conductance.
energies = [-2 * pot + 4. / 50. * pot * i for i in range(51)]
plot_conductance(syst, energies)
```

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

Technical details

In a lattice with more than one basis atom, you can always act either on all sublattices at the same time, or on a single sublattice only.

For example, you can add lattice points for all sublattices in the current example using:

syst[graphene.shape(...)] = ...

or just for a single sublattice:

syst[a.shape(...)] = ...

and the same of course with

*b*. Also, you can selectively remove points:del syst[graphene.shape(...)] del syst[a.shape(...)]

where the first line removes points in both sublattices, whereas the second line removes only points in one sublattice.