Re: [Kwant] Quantum Hall effect in graphene

30 Nov 2015

      Dear all,

For honeycomb lattice, is it still correct to use the hopping  -t *
np.exp(-1j * phi * (xt-xs) *(yt+ys)) in the presence of a magnetic field? I
am just wondering the vaildity of it because I cannot recover the
conductance steps in QHE.

Any suggestions will be appreciated. Thanks.

Best,
Johnny

On 29 November 2015 at 19:13, T.C. Wu  wrote:
...
Dear Anton,
Thanks for your prompt reply. However, how should I amend the code (which
works for square lattice in the example) in order to recover the
conductance plateaus? Thank you again.
Best,
Johnny
On 29 November 2015 at 18:35, Anton Akhmerov 
wrote:
...
Dear Johnny,
The error message you're seeing isn't lying. You probably have a lead
with zigzag boundary conditions that has a bad singularity at E=0.
Choosing even E=1e-4 would resolve the problem. And yes, you're right
about the spin density (but of course if you don't add any
spin-dependent terms, there won't be any spin physics happening.
Best regards,
Anton Akhmerov
...
Dear all,
I am trying to use kwant to reproduce the QHE in graphene. I combine the
example in qhe.py and graphene.py provided in the official website(see
...
code below).However, I got the following error when I plot out the
wavefunction at energy = 0:
RuntimeError: Numbers of left- and right-propagating modes differ,
On Sun, Nov 29, 2015 at 11:28 AM, T.C. Wu  wrote:
the
possibly
...
due to a numerical instability.
Moreover, if I don't choose energy = 0, say energy = 0.1, then the error
disappear. However, I cannot recover the plateaus in the plot of
conductance.
Lastly, it is correct to change t -> t*sigma_0 in order to extract the
distribution of spin(i.e. choosing the odd /even elements in the
wavefunction by [1::2]/[0::2]) in this case?
Thank you.
Best,
Johnny
Code:
# 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
#execfile('C:/Users/wtc/Google Drive/Linnian/plot_graphene.py')
from __future__ import division  # so that 1/2 == 0.5, and not 0
from math import pi, sqrt, tanh
import random
import kwant
import numpy as np
# For computing eigenvalues
import scipy.sparse.linalg as sla
# For plotting
from matplotlib import pyplot
# Define the graphene lattice
sin_30, cos_30 = (1 / 2, sqrt(3) / 2)
graphene = kwant.lattice.general([(1, 0), (sin_30, cos_30)],
                                 [(0, 0), (0, 1 / sqrt(3))])
a, b = graphene.sublattices
Bs = np.linspace(4,50,200)
t = -1
def make_system(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
        #return y > (abs(x)*np.tan(np.pi/3) -size) and y < size
    sys = 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)
def hopping2(site1, site2, phi = 0,salt=0):
        xt, yt = site1.pos
        xs, ys = site2.pos
        return -t * np.exp(-1j * phi * (xt-xs) *(yt+ys))
sys[graphene.shape(circle, (0, 0))] = 0
# 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))
    sys[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] =
hopping2
# Modify the scattering region
    # del sys[a(0, 0)]
    # del sys[a(5, 5)]
    # del sys[a(5,5)]
    # sys[a(-2, 1), b(2, 2)] = -1
#### 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))] = 0
    lead0[[kwant.builder.HoppingKind(*hopping) for hopping in
hoppings]] =
hopping2
# 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))] = 0
    lead1[[kwant.builder.HoppingKind(*hopping) for hopping in
hoppings]] =
hopping2
return sys, [lead0, lead1]
def compute_evs(sys):
    # Compute some eigenvalues of the closed system
    sparse_mat = sys.hamiltonian_submatrix(sparse=True)
evs = sla.eigs(sparse_mat, 2)[0]
    print evs.real
def plot_conductance(sys, energy):
    # Compute transmission as a function of energy
    data = []
    for phi in Bs:
        smatrix = kwant.smatrix(sys, energy,args =[phi,""])
        data.append(smatrix.transmission(0, 1))
pyplot.figure()
    pyplot.plot(Bs, 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()
def density(sys, energy, args, lead_nr):
    wf = kwant.wave_function(sys, energy, args)
    return (abs(wf(lead_nr))**2).sum(axis=0)
def main():
    pot = 0.1
    energy = 0.0
    sys, leads = make_system(pot=pot)
# 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(sys, site_color=family_colors, site_lw=0.1,
colorbar=False)
>
>     # Compute some eigenvalues.
>     compute_evs(sys.finalized())
>
>     # Attach the leads to the system.
>     for lead in leads:
>         sys.attach_lead(lead)
>
>     # Then, plot the system with leads.
>     kwant.plot(sys, site_color=family_colors, site_lw=0.1,
>                lead_site_lw=0, colorbar=False)
>
>     # Finalize the system.
>     sys = sys.finalized()
>
>     # Compute the band structure of lead 0.
>     momenta = [-4*pi + 0.02 * 4*pi * i for i in xrange(101)]
>     plot_bandstructure(sys.leads[0], momenta)
>
>
>     # plot wf
>     d = density(sys, energy, [1/40.0, ""], 0)
>     kwant.plotter.map(sys, d)
>     reciprocal_phis = np.linspace(4, 50, 200)
>     conductances = []
>     for phi in 1 / reciprocal_phis:
>         smatrix = kwant.smatrix(sys, energy, args=[phi, ""])
>         conductances.append(smatrix.transmission(0, 1)) # from lead 1 to
> lead 0
>     pyplot.plot(reciprocal_phis, conductances)
>     pyplot.show()
>     # Plot conductance.
>     # energies = [-2 * pot + 4. / 50. * pot * i for i in xrange(51)]
>     # plot_conductance(sys, 0)
>
>
> # Call the main function if the script gets executed (as opposed to
> imported).
> # See <http://docs.python.org/library/__main__.html>.
> if __name__ == '__main__':
>     main()
>