Hi Abbout, Thanks for the help.
-Leon From: Abbout Adel <abbout.a...@gmail.com<mailto:abbout.a...@gmail.com>> Date: Monday, October 17, 2016 at 1:59 PM To: "Leon Maurer (lmaurer)" <lmau...@sandia.gov<mailto:lmau...@sandia.gov>> Cc: "kwant-discuss@kwant-project.org<mailto:kwant-discuss@kwant-project.org>" <kwant-discuss@kwant-project.org<mailto:kwant-discuss@kwant-project.org>> Subject: [EXTERNAL] Re: [Kwant] a step tripping up Kwant Hi Leon, The upper and bottom limits of the conduction band are obtained from the relation of dispersion. E=V-2t *cos(k) (# in your case V=2t for left lead and V0 for the right lead ). For more details, you can look for example to : http://www-personal.umich.edu/~sunkai/teaching/Fall_2014/Chapter6.pdf Regards, Adel On Mon, Oct 17, 2016 at 10:34 PM, Maurer, Leon <lmau...@sandia.gov<mailto:lmau...@sandia.gov>> wrote: Hi Abbout, Is the upper bound on the energies in the leads documented somewhere? I guess it’s implicit in Sec. 2.4 of the tutorial, and now that you mention it, it makes sense given the periodic lattice. -Leon From: Abbout Adel <abbout.a...@gmail.com<mailto:abbout.a...@gmail.com>> Date: Monday, October 17, 2016 at 11:58 AM To: "Leon Maurer (lmaurer)" <lmau...@sandia.gov<mailto:lmau...@sandia.gov>> Cc: "kwant-discuss@kwant-project.org<mailto:kwant-discuss@kwant-project.org>" <kwant-discuss@kwant-project.org<mailto:kwant-discuss@kwant-project.org>> Subject: [EXTERNAL] Re: [Kwant] a step tripping up Kwant Dear Leon, the value of the parameter 't' in your program is around 16. this means that the conduction band for the left lead is band_l=[0, 4 t ]=[0,64] and the conduction band for the right lead is band_r=[V0, V0+4 t]=[100, 164] as you can notice there is no energy which conducts in both leads. In order to have a non zero transmission you need to use a value of V0<4 t. The result you are calling "exact" is valid for a continuous model: on a lattice, the dispersion relation is not quadratic. To compare, your result with the continuous limit, you need to choose V0<< 2t. For a "non uniform finite differences", you can look at the article [1] Hope that this helps. Adel [1]: http://scitation.aip.org/content/aip/journal/jap/68/8/10.1063/1.346245 On Mon, Oct 17, 2016 at 7:10 PM, Maurer, Leon <lmau...@sandia.gov<mailto:lmau...@sandia.gov>> wrote: 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 physics? (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.) Thanks. -Leon (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 plt.plot(x,U) # 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(i)]=U[i]+2*t sys[lat.neighbors()] = -t # set the finite-difference hopping parameters leftLead = kwant.Builder(kwant.TranslationalSymmetry((-a,))) # the lead to the left 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 right 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.figure() plt.plot(energies, data, energies, T) plt.legend(('numerical','exact'), loc=4) plt.xlabel("energy [V0]") plt.ylabel("Transmission") plt.show() return data, T # In[6]: stepNumerical, stepExact = plot_conductance(sys,linspace(1e-9,2*V0,201)) # In[7]: print(stepNumerical) -- Abbout Adel -- Abbout Adel
