@@ -1,179 +1,203 @@
# 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
+import _defs
from math import pi, sqrt, tanh
import kwant
# 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)
#HIDDEN_BEGIN_hnla
graphene = kwant.lattice.general([(1, 0), (sin_30, cos_30)],
[(0, 0), (0, 1 / sqrt(3))])
a, b = graphene.sublattices
#HIDDEN_END_hnla
#HIDDEN_BEGIN_shzy
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
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
#HIDDEN_END_shzy
# specify the hoppings of the graphene lattice in the
# format expected by builder.HoppingKind
#HIDDEN_BEGIN_hsmc
hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b))
#HIDDEN_END_hsmc
#HIDDEN_BEGIN_bfwb
syst[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = -1
#HIDDEN_END_bfwb
# Modify the scattering region
#HIDDEN_BEGIN_efut
del syst[a(0, 0)]
syst[a(-2, 1), b(2, 2)] = -1
#HIDDEN_END_efut
#### Define the leads. ####
#HIDDEN_BEGIN_aakh
# 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
#HIDDEN_END_aakh
#HIDDEN_BEGIN_kmmw
return syst, [lead0, lead1]
#HIDDEN_END_kmmw
#HIDDEN_BEGIN_zydk
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)
#HIDDEN_END_zydk
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()
+ fig = pyplot.figure()
pyplot.plot(energies, data)
- pyplot.xlabel("energy [t]")
- pyplot.ylabel("conductance [e^2/h]")
- pyplot.show()
+ pyplot.xlabel("energy [t]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("conductance [e^2/h]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("graphene_result." + extension, dpi=_defs.dpi)
def plot_bandstructure(flead, momenta):
bands = kwant.physics.Bands(flead)
energies = [bands(k) for k in momenta]
- pyplot.figure()
+ fig = pyplot.figure()
pyplot.plot(momenta, energies)
- pyplot.xlabel("momentum [(lattice constant)^-1]")
- pyplot.ylabel("energy [t]")
- pyplot.show()
+ pyplot.xlabel("momentum [(lattice constant)^-1]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.ylabel("energy [t]",
+ fontsize=_defs.mpl_label_size)
+ pyplot.setp(fig.get_axes()[0].get_xticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ pyplot.setp(fig.get_axes()[0].get_yticklabels(),
+ fontsize=_defs.mpl_tick_size)
+ fig.set_size_inches(_defs.mpl_width_in, _defs.mpl_width_in * 3. / 4.)
+ fig.subplots_adjust(left=0.15, right=0.95, top=0.95, bottom=0.15)
+ for extension in ('pdf', 'png'):
+ fig.savefig("graphene_bs." + extension, dpi=_defs.dpi)
#HIDDEN The part of the following code block which begins with family_colors
#HIDDEN is included verbatim in the tutorial text because nested code examples
#HIDDEN are not supported. Remember to update the tutorial text when you
#HIDDEN modify this block.
#HIDDEN_BEGIN_itkk
def main():
pot = 0.1
syst, 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(syst, site_color=family_colors, site_lw=0.1, colorbar=False)
+ size = (_defs.figwidth_in, _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_color=family_colors, site_lw=0.1, colorbar=False,
+ file="graphene_syst1." + extension,
+ fig_size=size, dpi=_defs.dpi)
#HIDDEN_END_itkk
# Compute some eigenvalues.
#HIDDEN_BEGIN_jmbi
compute_evs(syst.finalized())
#HIDDEN_END_jmbi
# Attach the leads to the system.
for lead in leads:
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)
+ size = (_defs.figwidth_in, 0.9 * _defs.figwidth_in)
+ for extension in ('pdf', 'png'):
+ kwant.plot(syst, site_color=family_colors, colorbar=False, site_lw=0.1,
+ file="graphene_syst2." + extension,
+ fig_size=size, dpi=_defs.dpi, lead_site_lw=0)
# Finalize the system.
syst = syst.finalized()
# Compute the band structure of lead 0.
momenta = [-pi + 0.02 * pi * i for i in range(101)]
plot_bandstructure(syst.leads[0], momenta)
# Plot conductance.
energies = [-2 * pot + 4. / 50. * pot * i for i in range(51)]
plot_conductance(syst, energies)
# 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()