Hello,
I have been trying to understand the conductance for Normal
metal-superconductor proximity system with repulsive interaction in the
normal layer numerically. The system I intend to solve is a semi-infinite
superconductor, located at x<0, in contact with a semi-infinite normal
metal with induced superconducting order parameter up to 0<x<d. Both
materials are infinite in the direction transverse to the SN interface. The
effect of repulsive interaction is that the sign of induced order parameter
in the normal layer is opposite to that in the bulk superconductor. The
source code for this system is following:
import kwant
from matplotlib import pyplot
def make_system(a=1, t=1.0, W=10, L=30, barrier=1.5, barrierpos=(14, 15),
mu=0.4, Delta_N=0.04, Delta_S=0.1, Deltapos_N=9, Deltapos_S=15):
lat_e = kwant.lattice.square(a, name='e')
lat_h = kwant.lattice.square(a, name='h')
sys = kwant.Builder()
#### Define the scattering region. ####
sys[(lat_e(x, y) for x in range(L) for y in range(W))] = 4 * t - mu
sys[(lat_h(x, y) for x in range(L) for y in range(W))] = mu - 4 * t
# hoppings for both electrons and holes
sys[lat_e.neighbors()] = -t
sys[lat_h.neighbors()] = t
for i in range(Deltapos_N, L):
for j in xrange(W):
if i < barrierpos[0]:
sys[lat_e(i, j), lat_h(i, j)] = -Delta_N
elif i in range(barrierpos[0], barrierpos[1]):
sys[lat_e(i,j)] = 4*t + barrier - mu
sys[lat_h(i,j)] = mu - 4*t - barrier
else:
sys[lat_e(i, j), lat_h(i, j)] = Delta_S
#### Define the leads. ####
# Symmetry for the left leads.
sym_left = kwant.TranslationalSymmetry((-a, 0))
# left electron lead
lead0 = kwant.Builder(sym_left)
lead0[(lat_e(0, j) for j in xrange(W))] = 4 * t - mu
lead0[lat_e.neighbors()] = -t
# left hole lead
lead1 = kwant.Builder(sym_left)
lead1[(lat_h(0, j) for j in xrange(W))] = mu - 4 * t
lead1[lat_h.neighbors()] = t
# Then the lead to the right
# this one is superconducting and thus is comprised of electrons
# AND holes
sym_right = kwant.TranslationalSymmetry((a, 0))
lead2 = kwant.Builder(sym_right)
lead2 += lead0
lead2 += lead1
lead2[((lat_e(0, j), lat_h(0, j)) for j in xrange(W))] = Delta_S
#### Attach the leads and return the system. ####
sys.attach_lead(lead0)
sys.attach_lead(lead1)
sys.attach_lead(lead2)
return sys
def plot_conductance(sys, energies):
# Compute conductance
data = []
for energy in energies:
smatrix = kwant.smatrix(sys, energy)
# Conductance is N - R_ee + R_he
data.append(smatrix.submatrix(0, 0).shape[0] -
smatrix.transmission(0, 0) +
smatrix.transmission(1, 0))
pyplot.figure()
pyplot.plot(energies, data)
pyplot.xlabel("energy [t]")
pyplot.ylabel("conductance [e^2/h]")
pyplot.show()
kwant.plotter.bands(sys.leads[0])
def main():
sys = make_system()
# Check that the system looks as intended.
kwant.plot(sys)
# Finalize the system.
sys = sys.finalized()
plot_conductance(sys, energies=[0.002 * i +0.00001 for i in xrange(100)])
if __name__ == '__main__':
main()
I expect a peak in the conductance at zero energy. Now the question I am
having here is that I am able to observe the peak when chemical potential
(\mu) is of the order of bulk superconducting order parameter (Delta_S)
which is unphysical. Also, I must have induced order parameter in the
normal layer (Delta_N) much smaller than bulk superconducting order
parameter. Can you suggest me something on that? I have analytical results
for this type of system and I just want agreement of the same from
numerics.
Thanks
Abhishek
--
--
Abhishek Kumar
Department of Physical Sciences
University of Florida, Gainesville
FL 32608
Alternate e-mail ID - kumarabhi(a)ufl.edu
Mobile - +1-3522831740
"Life isn't about how to survive the storm, but how to dance in the rain,"
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