Mailman 3 December 2015 - Kwant-discuss

plot eigenvectors of a closed system
by Weiyuan Tong Dec. 10, 2015

Dec. 10, 2015

Dear all， I note in Tutorial 2.4.1. and 2.7.1 that we can plot eigenvectors of a closed system using the code: def plot_data(sys, n): import scipy.linalg as la sys = sys.finalized() ham = sys.hamiltonian_submatrix() evecs = la.eigh(ham)[1] wf = abs(evecs[:, n])**2 kwant.plotter.map(sys, wf, oversampling=10, cmap='gist_heat_r') plot_data(sys, 225) I want to consider a system with spin, so the hopping is a matrix. How to change the the kwant code to plot the eigenvectors of system that spin is considered. Thanks in advances, Weiyuan

2 2

Declare hopping between different type of lattices
by Sudin Ganguly Dec. 9, 2015

Dec. 9, 2015

Dear Sir, If my query does not suit in this discussion, please forgive me. I wanted to create a Y-shaped 3 terminal structure. For that, I declared 3 different type of lattices, namely lat_a,lat_b and lat_c. lat_a: rectangular shape of width W lat_b: upper arm of Y of width W lat_c: lower arm of Y of width W I am not able to declare hopping between lat_a,lat_b and lat_a,lat_c. Is there any way to declare hopping between two different type of lattices? I would also like to know, is there any reference to create Y_shaped 3-terminal junction? I have appended my code in this mail. Please find the attachment below. With Regards, Sudin

2 3

Quantum Hall effect in graphene
by T.C. Wu Dec. 2, 2015

Dec. 2, 2015

Dear all, I am trying to use kwant to reproduce the QHE in graphene. I combine the example in qhe.py and graphene.py provided in the official website(see the code below).However, I got the following error when I plot out the wavefunction at energy = 0: RuntimeError: Numbers of left- and right-propagating modes differ, possibly due to a numerical instability. Moreover, if I don't choose energy = 0, say energy = 0.1, then the error disappear. However, I cannot recover the plateaus in the plot of conductance. Lastly, it is correct to change t -> t*sigma_0 in order to extract the distribution of spin(i.e. choosing the odd /even elements in the wavefunction by [1::2]/[0::2]) in this case? Thank you. Best, Johnny Code: # Tutorial 2.5. Beyond square lattices: graphene # ============================================== # # Physics background # ------------------ # Transport through a graphene quantum dot with a pn-junction # # Kwant features highlighted # -------------------------- # - Application of all the aspects of tutorials 1-3 to a more complicated # lattice, namely graphene #execfile('C:/Users/wtc/Google Drive/Linnian/plot_graphene.py') from __future__ import division # so that 1/2 == 0.5, and not 0 from math import pi, sqrt, tanh import random import kwant import numpy as np # For computing eigenvalues import scipy.sparse.linalg as sla # For plotting from matplotlib import pyplot # Define the graphene lattice sin_30, cos_30 = (1 / 2, sqrt(3) / 2) graphene = kwant.lattice.general([(1, 0), (sin_30, cos_30)], [(0, 0), (0, 1 / sqrt(3))]) a, b = graphene.sublattices Bs = np.linspace(4,50,200) t = -1 def make_system(r=10, w=2.0, pot=0.1): #### Define the scattering region. #### # circular scattering region def circle(pos): x, y = pos return x ** 2 + y ** 2 < r ** 2 #return y > (abs(x)*np.tan(np.pi/3) -size) and y < size sys = kwant.Builder() # w: width and pot: potential maximum of the p-n junction def potential(site): (x, y) = site.pos d = y * cos_30 + x * sin_30 return pot * tanh(d / w) def hopping2(site1, site2, phi = 0,salt=0): xt, yt = site1.pos xs, ys = site2.pos return -t * np.exp(-1j * phi * (xt-xs) *(yt+ys)) sys[graphene.shape(circle, (0, 0))] = 0 # specify the hoppings of the graphene lattice in the # format expected by builder.HoppingKind hoppings = (((0, 0), a, b), ((0, 1), a, b), ((-1, 1), a, b)) sys[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = hopping2 # Modify the scattering region # del sys[a(0, 0)] # del sys[a(5, 5)] # del sys[a(5,5)] # sys[a(-2, 1), b(2, 2)] = -1 #### Define the leads. #### # left lead sym0 = kwant.TranslationalSymmetry(graphene.vec((-1, 0))) def lead0_shape(pos): x, y = pos return (-0.4 * r < y < 0.4 * r) lead0 = kwant.Builder(sym0) lead0[graphene.shape(lead0_shape, (0, 0))] = 0 lead0[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = hopping2 # The second lead, going to the top right sym1 = kwant.TranslationalSymmetry(graphene.vec((0, 1))) def lead1_shape(pos): v = pos[1] * sin_30 - pos[0] * cos_30 return (-0.4 * r < v < 0.4 * r) lead1 = kwant.Builder(sym1) lead1[graphene.shape(lead1_shape, (0, 0))] = 0 lead1[[kwant.builder.HoppingKind(*hopping) for hopping in hoppings]] = hopping2 return sys, [lead0, lead1] def compute_evs(sys): # Compute some eigenvalues of the closed system sparse_mat = sys.hamiltonian_submatrix(sparse=True) evs = sla.eigs(sparse_mat, 2)[0] print evs.real def plot_conductance(sys, energy): # Compute transmission as a function of energy data = [] for phi in Bs: smatrix = kwant.smatrix(sys, energy,args =[phi,""]) data.append(smatrix.transmission(0, 1)) pyplot.figure() pyplot.plot(Bs, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() def plot_bandstructure(flead, momenta): bands = kwant.physics.Bands(flead) energies = [bands(k) for k in momenta] pyplot.figure() pyplot.plot(momenta, energies) pyplot.xlabel("momentum [(lattice constant)^-1]") pyplot.ylabel("energy [t]") pyplot.show() def density(sys, energy, args, lead_nr): wf = kwant.wave_function(sys, energy, args) return (abs(wf(lead_nr))**2).sum(axis=0) def main(): pot = 0.1 energy = 0.0 sys, leads = make_system(pot=pot) # To highlight the two sublattices of graphene, we plot one with # a filled, and the other one with an open circle: def family_colors(site): return 0 if site.family == a else 1 # Plot the closed system without leads. kwant.plot(sys, site_color=family_colors, site_lw=0.1, colorbar=False) # Compute some eigenvalues. compute_evs(sys.finalized()) # Attach the leads to the system. for lead in leads: sys.attach_lead(lead) # Then, plot the system with leads. kwant.plot(sys, site_color=family_colors, site_lw=0.1, lead_site_lw=0, colorbar=False) # Finalize the system. sys = sys.finalized() # Compute the band structure of lead 0. momenta = [-4*pi + 0.02 * 4*pi * i for i in xrange(101)] plot_bandstructure(sys.leads[0], momenta) # plot wf d = density(sys, energy, [1/40.0, ""], 0) kwant.plotter.map(sys, d) reciprocal_phis = np.linspace(4, 50, 200) conductances = [] for phi in 1 / reciprocal_phis: smatrix = kwant.smatrix(sys, energy, args=[phi, ""]) conductances.append(smatrix.transmission(0, 1)) # from lead 1 to lead 0 pyplot.plot(reciprocal_phis, conductances) pyplot.show() # Plot conductance. # energies = [-2 * pot + 4. / 50. * pot * i for i in xrange(51)] # plot_conductance(sys, 0) # Call the main function if the script gets executed (as opposed to imported). # See <http://docs.python.org/library/__main__.html>. if __name__ == '__main__': main()

3 5

Conductance, transmission coefficient and temperature
by L.M J Dec. 1, 2015

Dec. 1, 2015

Dear all I'm new to kwant and I'm trying to use kwant to replicate the result of zigzag graphene nanoribbons. I noticed the result get from "smatrix.transmission" is in the unit of e^2/h, which is the conductance. In some other software, the value to represent the transmission is called "transmission coefficient". I found in some of the papers( http://iopscience.iop.org/article/10.7567/JJAP.52.06GD05/meta), the conductance G can be changed to transmission coefficient by an intermediate function *Kn*. [image: Inline images 1] [image: Inline images 2] My question is: 1. Are the results I get from "smatrix.transmission" equivalent to the transmission coefficient? if not, how can I change the conductance to transmission coefficient? Can I change the conductance representation into transmission coefficient by the above equation? The reason I want to do this is that I can easily get the current/voltage curve by integrating the transmission coefficient. 2. Can I get transmission coefficient by intrinsic function already implemented in kwant? 3. I remember in some other quantum transport software, I need to set temperature for the transport calculation. But i notice I don't need to consider temperature in kwant. If I want to take temperature into account (fermi distribution), how can I do it?(Using the above equation?) 4. I calculate the zigzag graphene nanoribbon using kwant. I set onsite energy =0 and hopping =-2.74. I noticed my results are a bit wired. The major difference is I don't have an transmission peak in fermi energy. Is this because I didn't passivate my edge with hydrogen? Thanks for advance. PS: One thing I need to mention in my codes is: if I set energy equals to 0.0, "kwant.smatrix(sca, energy)" will return an error. So i replace 0.0 with 1e-5. My kwant calculation: [image: Inline images 5] reference [image: Inline images 4] (reference https://www.quantumwise.com/documents/tutorials/latest/BasicGrapheneTutoria… ). kwant codes is attached. #=================== import kwant from matplotlib import pyplot from math import sin, cos, tan, sqrt import numpy as np l_a=1.0 zgnr=kwant.lattice.general([(sqrt(3)*l_a,0),(0,3*l_a)], [(0,0),(0.5*sqrt(3)*l_a,l_a/2),(0.5*sqrt(3)*l_a,3*l_a/2),(0,2*l_a)]) a,b,c,d=zgnr.sublattices ld=kwant.TranslationalSymmetry(zgnr.vec((-1,0))) rd=kwant.TranslationalSymmetry(zgnr.vec((1,0))) h=4 w=h def shape1(pos): x,y=pos return (-h+2<y<h and abs(x)<w) left_lead=kwant.Builder(ld) left_lead[zgnr.shape(shape1,(0,0))]=0 left_lead[[kwant.builder.HoppingKind((0,0),a,b)]]=-2.74 left_lead[[kwant.builder.HoppingKind((0,0),b,c)]]=-2.74 left_lead[[kwant.builder.HoppingKind((0,0),c,d)]]=-2.74 left_lead[[kwant.builder.HoppingKind((-1,0),b,a)]]=-2.74 left_lead[[kwant.builder.HoppingKind((-1,0),c,d)]]=-2.74 left_lead[[kwant.builder.HoppingKind((0,1),a,d)]]=-2.74 kwant.plot(left_lead); right_lead=kwant.Builder(rd) right_lead[zgnr.shape(shape1,(0,0))]=0 right_lead[[kwant.builder.HoppingKind((0,0),a,b)]]=-2.74 right_lead[[kwant.builder.HoppingKind((0,0),b,c)]]=-2.74 right_lead[[kwant.builder.HoppingKind((0,0),c,d)]]=-2.74 right_lead[[kwant.builder.HoppingKind((-1,0),b,a)]]=-2.74 right_lead[[kwant.builder.HoppingKind((-1,0),c,d)]]=-2.74 right_lead[[kwant.builder.HoppingKind((0,1),a,d)]]=-2.74 sca=kwant.Builder() sca[zgnr.shape(shape1,(0,0))]=0 sca[[kwant.builder.HoppingKind((0,0),a,b)]]=-2.74 sca[[kwant.builder.HoppingKind((0,0),b,c)]]=-2.74 sca[[kwant.builder.HoppingKind((0,0),c,d)]]=-2.74 sca[[kwant.builder.HoppingKind((-1,0),b,a)]]=-2.74 sca[[kwant.builder.HoppingKind((-1,0),c,d)]]=-2.74 sca[[kwant.builder.HoppingKind((0,1),a,d)]]=-2.74 sca.attach_lead(left_lead) sca.attach_lead(right_lead) sca=sca.finalized() kwant.plot(sca); #bands=kwant.physics.Bands(left_lead); #mom=np.linspace(-np.pi,np.pi,101) #ee=[bands(k) for k in mom] #pyplot.plot(mom, ee); #pyplot.grid('on') #pyplot.ylim(-2.2,2.2) momenta = [-np.pi + 0.02 * np.pi * i for i in xrange(101)] bands = kwant.physics.Bands(sca.leads[0]) ee = [bands(k) for k in momenta] pyplot.plot(momenta, ee) energies = [] data = [] for ie in xrange(501): energy = ie * 0.01-2.5 if energy ==0.0: energy = 1e-5 # compute the scattering matrix at a given energy smatrix = kwant.smatrix(sca, energy) # compute the transmission probability from lead 0 to # lead 1 energies.append(energy) data.append(smatrix.transmission(1, 0)) pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() #===========================

1 0