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@ufl.edu

Mobile - +1-3522831740

"Life isn't about how to survive the storm, but how to dance in the rain," 

                                                                                             - Unknown