I searched a lot and got the answer!
Here it is some links that helped me
Some exemples links:
https://kwant-project.org/doc/1/tutorial/spin_potential_shape.html?highlight=spin
https://kwant-project.org/doc/1/_downloads/spin_orbit.py
https://kwant-project.org/doc/1/tutorial/superconductors.html?highlight=norbs
Some forum threads links:
https://www.mail-archive.com/kwant-discuss@kwant-project.org/msg01353.html
https://kwant-discuss.kwant-project.narkive.com/96MScaQw/kwant-spin-currents-using-greens-functions#post4
Summarizing: To obtain spin up-up, up-down, down-up and down-down
transmissions, it is required use norbs = 2 in a lattice construction and make
the scattering matrix ordered.
Attention in lat and lead declaration
Code exemple:
import kwant
import numpy as np
# For plotting
from matplotlib import pyplot
# For matrix support
import tinyarray
# define Pauli-matrices for convenience
sigma_0 = tinyarray.array([[1, 0], [0, 1]])
sigma_x = tinyarray.array([[0, 1], [1, 0]])
sigma_y = tinyarray.array([[0, -1j], [1j, 0]])
sigma_z = tinyarray.array([[1, 0], [0, -1]])
def make_system(t=1.0, alpha=0.5, e_z=0.08, W=10, L=30):
# Start with an empty tight-binding system and a single square lattice.
# `a` is the lattice constant (by default set to 1 for simplicity).
lat = kwant.lattice.square(norbs = 2) # these orbitals will be one to spin
up and one to spin down
syst = kwant.Builder()
#### Define the scattering region. ####
syst[(lat(x, y) for x in range(L) for y in range(W))] = \
4 * t * sigma_0 + e_z * sigma_z
# hoppings in x-direction
syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
-t * sigma_0 + 1j * alpha * sigma_y / 2
# hoppings in y-directions
syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
-t * sigma_0 - 1j * alpha * sigma_x / 2
#### Define the left lead. ####
lead = kwant.Builder(kwant.TranslationalSymmetry((-1, 0)),
conservation_law=-sigma_z) # this sorts the blocks: "up" modes come first, and
then the "down"
lead[(lat(0, j) for j in range(W))] = 4 * t * sigma_0 + e_z * sigma_z
# hoppings in x-direction
lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = \
-t * sigma_0 + 1j * alpha * sigma_y / 2
# hoppings in y-directions
lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \
-t * sigma_0 - 1j * alpha * sigma_x / 2
#### Attach the leads and return the finalized system. ####
syst.attach_lead(lead)
syst.attach_lead(lead.reversed())
return syst
def plot_conductance(syst, energies):
# Compute conductance
data = [] #total transmission
data_uu = [] #up-up transmission
data_ud = [] #up-down transmission
data_du = [] #down-up transmission
data_du = [] #down-up transmission
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
t = np.matrix(smat.submatrix(1, 0)) #transmission block
tt = np.matmul(t, t.getH()) #t*t_dagger matrix in case of especific
cases, i prefer work with this form
data.append(smatrix.transmission(1, 0)) # transmission lead 0 to lead 1
data_uu.append(smatrix.transmission((1,0), (0,0))) # transmission lead
0 up to lead 1 up
data_ud.append(smatrix.transmission((1,1), (0,0))) # transmission lead
0 up to lead 1 down
data_du.append(smatrix.transmission((1,0), (0,1))) # transmission lead
0 down to lead 1 up
data_dd.append(smatrix.transmission((1,0), (0,0))) # transmission lead
0 down to lead 1 down
pyplot.figure()
pyplot.plot(energies, data)
pyplot.xlabel("energy [t]")
pyplot.ylabel("conductance [e^2/h]")
pyplot.show()
pyplot.figure()
pyplot.plot(energies, data_uu)
pyplot.xlabel("energy [t]")
pyplot.ylabel("up-up conductance [e^2/h]")
pyplot.show()
pyplot.figure()
pyplot.plot(energies, data_ud)
pyplot.xlabel("energy [t]")
pyplot.ylabel("up-down conductance [e^2/h]")
pyplot.show()
pyplot.figure()
pyplot.plot(energies, data_du)
pyplot.xlabel("energy [t]")
pyplot.ylabel("down-up conductance [e^2/h]")
pyplot.show()
pyplot.figure()
pyplot.plot(energies, data_dd)
pyplot.xlabel("energy [t]")
pyplot.ylabel("down-down conductance [e^2/h]")
pyplot.show()
def main():
syst = make_system()
# Check that the system looks as intended.
kwant.plot(syst)
# Finalize the system.
syst = syst.finalized()
# We should see non-monotonic conductance steps.
plot_conductance(syst, energies=[0.01 * i - 0.3 for i in range(100)])
# 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()
Hope it helps!
