Hello everyone,

I’ve been playing around with Kwant and come across some situations where the 
transmission between two leads is identically equal to zero when I wouldn’t 
expect that result.

I’ve come up with a simple working example: a 1D step function with a step 
height that’s large relative to the lattice spacing (code below), altho this 
problem seems to sometimes crop up in other, somewhat less extreme situations.

I understand that the numerical result should become less accurate as the 
step-height-to-lattice-spacing ratio increases, but why does the transmission 
become identically equal to zero at some point? Are there well-defined 
conditions for when this happens? Is there some way to know that the 
transmission is zero because of numerical issues rather than the underlying 

(Ultimately, I’m interested in modeling some systems where the potential mostly 
varies gradually but has a few small regions with abrupt changes in potential. 
Moving to a finer mesh (smaller lattice constant) everywhere is 
cost-prohibitive. Having some tool to easily refine the mesh in a region would 
be very useful.)



(Below code taken from jupyter notebook.)

# In[1]:

get_ipython().magic('load_ext autoreload')
get_ipython().magic('autoreload 2')
from numpy import *
import matplotlib.pyplot as plt
get_ipython().magic('matplotlib inline')
import tqdm
import kwant

# In[2]:

m0 = 9.10938215e-31 # Electron mass, [kg]
hbar = 1.054571726e-34 # hbar in [J] [s]
q = 1.602176565e-19 # Elementary charge, [C]
mt = 0.19
ml = 0.92

m = mt*m0

# In[3]:

V0 = 100 # step height
x = linspace(0,100,30) # thirty grid points
U = zeros_like(x)
U[len(x)//2:] = V0

# In[4]:

a = x[1]-x[0] # grid spacing [nm]
t = hbar**2/(2.*m*(a*1e-9)**2)/q*1e3 #hopping parameter [meV]

lat = kwant.lattice.chain(a) # Set up the transport simulation on a 1D latice
sys = kwant.Builder() # initialize the transport simulation
for i in range(len(U)): # populate based on the potential landscape

sys[lat.neighbors()] = -t # set the finite-difference hopping parameters
leftLead = kwant.Builder(kwant.TranslationalSymmetry((-a,))) # the lead to the 
leftLead[lat(0)] = 2*t + U[0]
leftLead[lat.neighbors()] = -t
sys.attach_lead(leftLead) # attach it

rightLead = kwant.Builder(kwant.TranslationalSymmetry((a,))) # the lead to the 
rightLead[lat(0)] = 2*t + U[-1]
rightLead[lat.neighbors()] = -t
sys.attach_lead(rightLead) # attach it
sys = sys.finalized()

# In[5]:

def plot_conductance(sys, energies):
    # Compute transmission numerically
    data = []
    for energy in tqdm.tqdm(energies,leave=True):
        smatrix = kwant.smatrix(sys, energy)
        data.append(smatrix.transmission(1, 0))

    # Compute exact conductance
    k1 = sqrt(2*m*energies/hbar**2)
    k2 = sqrt(2*m*(energies - V0)/hbar**2)
    T = 4*k1*k2/(k1+k2)**2
    T[energies <= V0] = 0

    plt.plot(energies, data, energies, T)
    plt.legend(('numerical','exact'), loc=4)
    plt.xlabel("energy [V0]")
    return data, T

# In[6]:

stepNumerical, stepExact = plot_conductance(sys,linspace(1e-9,2*V0,201))

# In[7]:


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