2.4. Beyond transport: Band structure and closed systems

Band structure calculations

See also

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See also

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

# Tutorial 2.4.1. Band structure calculations
# ===========================================
#
# Physics background
# ------------------
#  band structure of a simple quantum wire in tight-binding approximation
#
# Kwant features highlighted
# --------------------------
#  - Computing the band structure of a finalized lead.

import kwant

# For plotting
from matplotlib import pyplot

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')

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:

def make_lead(a=1, t=1.0, W=10):
    # Start with an empty lead with a single square lattice
    lat = kwant.lattice.square(a, norbs=1)

    sym_lead = kwant.TranslationalSymmetry((-a, 0))
    lead = kwant.Builder(sym_lead)

    # build up one unit cell of the lead, and add the hoppings
    # to the next unit cell
    for j in range(W):
        lead[lat(0, j)] = 4 * t

        if j > 0:
            lead[lat(0, j), lat(0, j - 1)] = -t

        lead[lat(1, j), lat(0, j)] = -t

    return lead

“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 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 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 Bands. For now we just plot the bandstructure:

def main():
    lead = make_lead().finalized()
    kwant.plotter.bands(lead, show=False)
    pyplot.xlabel("momentum [(lattice constant)^-1]")
    pyplot.ylabel("energy [t]")
    pyplot.show()

This gives the result:

# Call the main function if the script gets executed (as opposed to imported).
# See <https://docs.python.org/library/__main__.html>.
if __name__ == '__main__':
    main()

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

See also

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

# Tutorial 2.4.2. Closed systems
# ==============================
#
# Physics background
# ------------------
#  Fock-darwin spectrum of a quantum dot (energy spectrum in
#  as a function of a magnetic field)
#
# Kwant features highlighted
# --------------------------
#  - Use of `hamiltonian_submatrix` in order to obtain a Hamiltonian
#    matrix.

from cmath import exp
import numpy as np
from matplotlib import pyplot
import kwant

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')

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:

# For eigenvalue computation
import scipy.sparse.linalg as sla

We set up the system using the shape-function as in Nontrivial shapes, but do not add any leads:

a = 1
t = 1.0
r = 10

def make_system(a=1, t=1.0, r=10):

    lat = kwant.lattice.square(a, norbs=1)

    syst = kwant.Builder()

    # Define the quantum dot
    def circle(pos):
        (x, y) = pos
        rsq = x ** 2 + y ** 2
        return rsq < r ** 2

    def hopx(site1, site2, B):
        # The magnetic field is controlled by the parameter B
        y = site1.pos[1]
        return -t * exp(-1j * B * y)

    syst[lat.shape(circle, (0, 0))] = 4 * t
    # hoppings in x-direction
    syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = hopx
    # hoppings in y-directions
    syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t

    # It's a closed system for a change, so no leads
    return syst

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 hamiltonian_submatrix:

def plot_spectrum(syst, Bfields):

    energies = []
    for B in Bfields:
        # Obtain the Hamiltonian as a sparse matrix
        ham_mat = syst.hamiltonian_submatrix(params=dict(B=B), sparse=True)

        # we only calculate the 15 lowest eigenvalues
        ev = sla.eigsh(ham_mat.tocsc(), k=15, sigma=0,
                       return_eigenvectors=False)

        energies.append(ev)

    pyplot.figure()
    pyplot.plot(Bfields, energies)
    pyplot.xlabel("magnetic field [arbitrary units]")
    pyplot.ylabel("energy [t]")
    pyplot.show()

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

syst = make_system()

syst = syst.finalized()

# We should observe energy levels that flow towards Landau
# level energies with increasing magnetic field.
plot_spectrum(syst, [iB * 0.002 for iB in range(100)])

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 map:

def sorted_eigs(ev):
    evals, evecs = ev
    evals, evecs = map(np.array, zip(*sorted(zip(evals, evecs.transpose()))))
    return evals, evecs.transpose()

def plot_wave_function(syst, B=0.001):
    # Calculate the wave functions in the system.
    ham_mat = syst.hamiltonian_submatrix(sparse=True, params=dict(B=B))
    evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))

    # Plot the probability density of the 10th eigenmode.
    kwant.plotter.map(syst, np.abs(evecs[:, 9])**2,
                      colorbar=False, oversampling=1)

syst = make_system(r=30)

# Plot an eigenmode of a circular dot. Here we create a larger system for
# better spatial resolution.
syst = make_system(r=30).finalized()
plot_wave_function(syst);

The last two arguments to 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:

def plot_current(syst, B=0.001):
    # Calculate the wave functions in the system.
    ham_mat = syst.hamiltonian_submatrix(sparse=True, params=dict(B=B))
    evals, evecs = sorted_eigs(sla.eigsh(ham_mat.tocsc(), k=20, sigma=0))

    # Calculate and plot the local current of the 10th eigenmode.
    J = kwant.operator.Current(syst)
    current = J(evecs[:, 9], params=dict(B=B))
    kwant.plotter.current(syst, current, colorbar=False)

plot_current(syst);

Technical details :class: dropdown note

• hamiltonian_submatrix can also return a sparse matrix, if the optional argument sparse=True. The sparse matrix is in SciPy’s scipy.sparse.coo_matrix format, which can be easily be converted to various other sparse matrix formats (see SciPy’s documentation).

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”.