[Kwant] Quantum Hall Effect in a graphene Hall Bar

29 Jul 2021


      Hello, I am trying to simulate the quantum hall effect in a hall bar made of a honeycomb (graphene) lattice. I modified the code submitted by C. Groth as a response to a 2015 thread (link below). As soon as I add the vertical leads, I cant seem to get the expected phenomenon for both longitudinal and hall resistances. Here is my code, if anyone spots an error or a solution, do let me know!

Sid Sule
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from cmath import exp
import numpy
from matplotlib import pyplot

import kwant
from kwant.digest import gauss


def hopping(sitei, sitej, phi, salt):
    xi, yi = sitei.pos
    xj, yj = sitej.pos
    return -exp(-0.5j * phi * (xi - xj) * (yi + yj))


def onsite(site, phi, salt):
    return 0.05 * gauss(repr(site), salt)


def make_system(L=20, W=30):

    def central_region(pos):
        x, y = pos
        return abs(x) < (W/2) and abs(y) < (L/2)

    lat = kwant.lattice.honeycomb()
    sys = kwant.Builder()

    sys[lat.shape(central_region, (0, 0))] = onsite
    sys[lat.neighbors()] = hopping

    vertices = numpy.array([[-15, -6, 5, 10],
                            [-15, -6, -10, -5],
                            [6, 15, 5, 10],
                            [6, 15, -10, -5],
                            [-2, 2, 5, 10],
                            [-2, 2, -10, -5]])

    for v in vertices:

        def in_hole(site):

            x, y = site.pos
            return (v[0] <= x <= v[1]) and (v[2] <= y <= v[3])

        for site in filter(in_hole, list(sys.sites())):

            del sys[site]

    sym = kwant.TranslationalSymmetry((-1, 0))
    lead = kwant.Builder(sym)

    lead[lat.shape(lambda s: abs(s[1]) < (L/4), (0, 0))] = 0
    lead[lat.neighbors()] = hopping

    sys.attach_lead(lead)
    sys.attach_lead(lead.reversed())

    edges = numpy.array([[-6, -2, -4],
                         [2, 6, 4]])

    for e in edges:

        sym = kwant.TranslationalSymmetry([0, numpy.sqrt(3)])
        lead = kwant.Builder(sym)

        def lead_region(pos):

            x, y = pos
            return (e[0] < x < e[1])

        lead[lat.shape(lead_region, (e[2], 0))] = 0
        lead[lat.neighbors()] = hopping

        sys.attach_lead(lead)
        sys.attach_lead(lead.reversed())

        for x in lead.sites():
            x.family.norbs = None

    return sys.finalized()


sys = make_system()
kwant.plot(sys, site_lw=0.1, lead_site_lw=0, colorbar=False, show=False)

energy = 0.35
reciprocal_phis = numpy.linspace(4, 50, 20)

current = numpy.array([-1, 1, 0, 0, 0, 0])

r_long = []
r_hall = []
for phi in 1 / reciprocal_phis:

    params = {"phi": phi, "salt": ""}

    smatrix = kwant.smatrix(sys, energy, params=params)
    cond_mat = smatrix.conductance_matrix()

    # SOLVES FOR V IN I = G*V
    voltage = numpy.linalg.solve(cond_mat, current)

    # DIVIDE BY 2 FOR  V = I*R, I = 2
    r_long.append(abs(voltage[2]-voltage[4]) / 2)
    r_hall.append((voltage[2] - voltage[3]) / 2)

fig, ax = pyplot.subplots(1, 1)
ax.plot(reciprocal_phis, r_long)
ax.plot(reciprocal_phis, r_hall)

fig.show()

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Link of original thread : https://mail.python.org/archives/list/kwant-discuss@python.org/thread/D3MTMB...

[Kwant] Quantum Hall Effect in a graphene Hall Bar

sidsule3000＠hotmail.com