2.5. Beyond square lattices: graphene#

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.