Dear all,
in the tutorial
https://kwant-project.org/doc/1/tutorial/superconductors
one calculates the conductance from left lead to right lead, G_10, as
G_00 = e^2/h (N-R_ee+R_he)
assuming that, due to the current conservation,
G_10 = G_00.
My question is, what happens if we detach the superconducting lead L1.
Intuitively, I would expect
G_00 = 0. But using the formula above one obtains non-zero values, and making
the finite superconducting
region (the part of the scattering region) large enough, the subgap transport
is exactly as if
the lead L1 was attached. This means that the current conservation is violated,
and something is wrong.
Below is slightly modified code from the tutorial to generate a plot comparing
G_00 for a system with and
without lead L1. You can see, that if parameter L (the superconducting part of
the scattering region) is large
enough, the subgap transport is the same whether L1 is attached or not.
Why is this happening, can someone point out the error?
Best,
Tibor
import kwant
import tinyarray
import numpy as np
%matplotlib inline
from matplotlib import pyplot as plt
tau_x = tinyarray.array([[0, 1], [1, 0]])
tau_y = tinyarray.array([[0, -1j], [1j, 0]])
tau_z = tinyarray.array([[1, 0], [0, -1]])
L=80
barrier=1.5
W=10
def make_system(a=1, W=W, L=L, barrier=barrier, barrierpos=(3, 4),
mu=0.4, Delta=0.1, Deltapos=4, t=1.0, phs=True):
lat = kwant.lattice.square(norbs=2)
syst = kwant.Builder()
syst[(lat(x, y) for x in range(0,Deltapos) for y in range(W))] = (4 * t -
mu) * tau_z
syst[(lat(x, y) for x in range(Deltapos, L) for y in range(W))] = (4 * t -
mu) * tau_z + Delta * tau_x
# The tunnel barrier
syst[(lat(x, y) for x in range(barrierpos[0], barrierpos[1])
for y in range(W))] = (4 * t + barrier - mu) * tau_z
# Hoppings
syst[lat.neighbors()] = -t * tau_z
sym_left = kwant.TranslationalSymmetry((-a, 0))
lead0 = kwant.Builder(sym_left, conservation_law=-tau_z,
particle_hole=tau_y)
lead0[(lat(0, j) for j in range(W))] = (4 * t - mu) * tau_z
lead0[lat.neighbors()] = -t * tau_z
sym_right = kwant.TranslationalSymmetry((a, 0))
lead1 = kwant.Builder(sym_right)
lead1[(lat(0, j) for j in range(W))] = (4 * t - mu) * tau_z + Delta * tau_x
lead1[lat.neighbors()] = -t * tau_z
syst.attach_lead(lead0)
syst.attach_lead(lead1)
return syst
syst = make_system()
kwant.plot(syst)
syst = syst.finalized()
data = []
energies=[0.002 * i for i in range(0,100)]
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
# Conductance is N - R_ee + R_he
data.append(smatrix.submatrix((0, 0), (0, 0)).shape[0] -
smatrix.transmission((0, 0), (0, 0)) +
smatrix.transmission((0, 1), (0, 0)))
syst = make_system()
del syst.leads[-1]
# Check that the system looks as intended.
kwant.plot(syst)
# Finalize the system.
syst = syst.finalized()
data2 = []
for energy in energies:
smatrix = kwant.smatrix(syst, energy)
# Conductance is N - R_ee + R_he
data2.append(smatrix.submatrix((0, 0), (0, 0)).shape[0] -
smatrix.transmission((0, 0), (0, 0)) +
smatrix.transmission((0, 1), (0, 0)))
fig , (ax0) = plt.subplots(1, 1, sharey = True, figsize = [5, 5])
ax0.plot(energies, data, label = 'with lead 1')
ax0.plot(energies, data2, label = 'without lead 1')
ax0.set_xlabel("energy [t]")
ax0.set_ylabel("conductance [e^2/h]")
ax0.set_ylim(0,3.5)
ax0.legend()
fig.savefig('NS_conductance.png')
