See also
The complete source code of this example can be found in
tutorial/band_structure.py
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)
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 xrange(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:
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.
See also
The complete source code of this example can be found in
tutorial/closed_system.py
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:
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:
lat = kwant.lattice.square(a)
sys = 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=0):
# The magnetic field is controlled by the parameter B
y = site1.pos[1]
return -t * exp(-1j * B * y)
sys[lat.shape(circle, (0, 0))] = 4 * t
# hoppings in x-direction
sys[kwant.builder.HoppingKind((1, 0), lat, lat)] = hopx
# hoppings in y-directions
sys[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t
# It's a closed system for a change, so no leads
return sys
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
and 
.)
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(sys, Bfields):
# In the following, we compute the spectrum of the quantum dot
# using dense matrix methods. This works in this toy example, as
# the system is tiny. In a real example, one would want to use
# sparse matrix methods
energies = []
for B in Bfields:
# Obtain the Hamiltonian as a dense matrix
ham_mat = sys.hamiltonian_submatrix(args=[B], sparse=True)
# we only calculate the 15 lowest eigenvalues
ev = sla.eigsh(ham_mat, k=15, which='SM', 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:
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 plot_wave_function(sys):
# Calculate the wave functions in the system.
ham_mat = sys.hamiltonian_submatrix(sparse=True)
evecs = sla.eigsh(ham_mat, k=20, which='SM')[1]
# Plot the probability density of the 10th eigenmode.
kwant.plotter.map(sys, np.abs(evecs[:, 9])**2,
colorbar=False, oversampling=1)
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.
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”. |