Mailman 3 November 2015 - Kwant-discuss

How the spin is positioned in the Smatrix?
by Qingtian Zhang 07 Mar '22

07 Mar '22

Dear all, I am trying square lattice with spin-orbit coupling and Zeeman splititng just like the example of Tutorial 2.3.1. Now I want to get the spin-dependent conductance：Guu,Gud,Gdu,and Gdd, where "u" indicates up spin, d indicates down spin. We can have the scattering matrix through Smatrix=kwant.smatrix(sys, energy), but how the spin is positioned in the scattering matrix? I have checked some simple cases, it seems the Smatrix is organized like this: | r t | S= | t' r' | but how the spin is included? Thanks in advance. Qingtiian Zhang The code is: import kwant from matplotlib import pyplot from numpy import matrix import tinyarray sigma_0 = tinyarray.array([[1, 0], [0, 1]]) sigma_x = tinyarray.array([[0, 1], [1, 0]]) sigma_y = tinyarray.array([[0, -1j], [1j, 0]]) sigma_z = tinyarray.array([[1, 0], [0, -1]]) sys=kwant.Builder() lat=kwant.lattice.square() a=1 t=1.0 alpha=0.5 e_z=0.08 W=2 L=2 sys[(lat(x, y) for x in range(L) for y in range(W))] = \ 4 * t * sigma_0 + e_z * sigma_z # hoppings in x-direction sys[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_y # hoppings in y-directions sys[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_x #### Define the left lead. #### lead = kwant.Builder(kwant.TranslationalSymmetry((-a, 0))) lead[(lat(0, j) for j in xrange(W))] = 4 * t * sigma_0 + e_z * sigma_z # hoppings in x-direction lead[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_y # hoppings in y-directions lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_x #### Attach the leads and return the finalized system. #### sys.attach_lead(lead) sys.attach_lead(lead.reversed()) kwant.plot(sys) sys=sys.finalized() print "Hamiltonian=" print sys.hamiltonian_submatrix() Smatrix=kwant.smatrix(sys, energy=3.) print "The scattering matrix=" print matrix.round((Smatrix.data),2) print "T=",(Smatrix.transmission(1, 0))

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How to caculate hall resistance
by ZHANG Bing 12 Sep '21

12 Sep '21

Dear Sir, I am a PhD student of Hong Kong University of Science and Technology. I want to use KWANT to caculate Hall resistance of a Hall bar structure.We can get the conductance between 6 electrodes, but how to get hall resistance? Can you give me some help? Thank you very much. Best Regards, Zhang Bing

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New instructions for Windows installation
by Michael Wimmer 15 Jan '16

15 Jan '16

Dear kwant users, we have posted new instructions for installing kwant on Windows on the kwant website: http://www.kwant-project.org/install The old instructions do not work any more, as the format of the python packages on Christoph Gohlke's webpage have changed. Best regards, the kwant team

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Quantum Hall effect in graphene
by T.C. Wu 02 Dec '15

02 Dec '15

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()

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High symmetry band structure calculation
by liming jiang 30 Nov '15

30 Nov '15

Dear all Is there an easy way to produce band structure plot over certain high symmetry point in brillouin zone? In order to calculate the bandstructure of certain lattice, graphene for example, One has to define a lead with periodicity in both a and b direction. >ge=kwant.lattice.general([(1.0,0),(sin_30,cos_30)], [(0,0),(0,1.0/sqrt(3))]) >lead=kwant.TranslationalSymmetry(ge.vec((-1,0)),ge.vec((0,1))) But the kwant doesn't allows me to finalize the system. Did I made a mistake or kwant currently doesn't support the brillouin zone bandstructure calculation Thanks for advance. Best regards, Liming

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Define an infinite system?
by L.M J 30 Nov '15

30 Nov '15

I was trying to calculate the arm-chair graphene nano ribbon band structure. I used translational symmetry to define the lead to be infinite, but I still got an error saying "TypeError: Expecting an instance of InfiniteSystem." I can't identify what's problem here. can anyone help? Thanks for advance. code: ================================ l_a=1.0 agnr=kwant.lattice.general([(3*l_a,0),(0,sqrt(3)*l_a)], [(0,0),(0.5*l_a,sqrt(3)*l_a/2),(1.5*l_a,sqrt(3)*l_a/2),(2*l_a,0)]) a,b,c,d=agnr.sublattices print agnr.vec((-1,0)) ld=kwant.TranslationalSymmetry(agnr.vec((-1,0))) h=10 w=h def shape1(pos): x,y=pos return (abs(y)<h and abs(x)<w) left_lead=kwant.Builder(ld) left_lead[agnr.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),d,a)]]=-2.74 left_lead[[kwant.builder.HoppingKind((0,1),a,b)]]=-2.74 left_lead[[kwant.builder.HoppingKind((0,1),d,c)]]=-2.74 kwant.plot(left_lead); left_lead.finalized(); 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)

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Re: [Kwant] Distribution of spin in graphene lattice
by T.C. Wu 27 Nov '15

27 Nov '15

Dear Anton, Thanks for your reply. If I don't include the spin-orbital coupling term and just want to see how are the spin up /down components of the wavefunction distributed on the graphene surface, is it possible to do it by 1. change the n.n. hopping from -t to -t*sigma0 2. just plot out the even/odd index of the wavefunction (just like in https://mailman-mail5.webfaction.com/pipermail/kwant-discuss/2013-September… )? -Originally, I have 86 wavefunctions in total (since there are 86 eigenvalues). -By using the above approach, I get 172 wavefunctions (now there are 172 eigenvalues, ). -But when I sum the ground state wavefunction together, I cannot recover the original ground state wavefunction. Part of my code is as follows. ####### Case 1: hopping = -t * sigma_0 ######## iplot = 1 evecs = la.eigh(ham)[1] wf_up = abs(evecs[::2, 2*iplot])**2 wf_down = abs(evecs[1::2,2*iplot])**2 Note: Now the number of wavefunction obtained is doubled (say 2n in total). But the 1st and the 2nd one seems to be the same, so I just pick the (2*i)th wavefunction wf = wf_up + wf_down def site_size_sum(i): return 1* wf[i] / wf.max() def site_size_up(i): return 1* wf_up[i] / wf.max() def site_size_down(i): return 1* wf_down[i] / wf.max() kwant.plot(sys, site_size=site_size_up, site_color=(0, 0, 1, 0.3),hop_lw=0.1) kwant.plot(sys, site_size=site_size_down, site_color=(0, 0, 1, 0.3),hop_lw=0.1) kwant.plot(sys, site_size=site_size_sum, site_color=(0, 0, 1, 0.3),hop_lw=0.1) ####### Case 2: hopping = -t ########## wf = abs(evecs[:, iplot])**2 def site_size(i): return 1* wf[i] / wf.max() kwant.plot(sys, site_size=site_size, site_color=(0, 0, 1, 0.3),hop_lw=0.1) I noticed that the plot of site_size_sum in Case 1 is not the same as Case 2. What should be correct way to do it? Any hint for the mistakes is appreciated. Thanks. Best, Johnny On 26 November 2015 at 19:01, Anton Akhmerov <anton.akhmerov(a)gmail.com> wrote: > Dear Johnny, > > Sorry for a delayed reply. Please send the email to the mailing list > and not only to me personally (that way someone else may reply if I'm > slow, and others can benefit from the discussion). > > Best, > Anton > > On Tue, Nov 24, 2015 at 10:54 AM, T.C. Wu <tcwu(a)connect.ust.hk> wrote: > > Dear Anton, > > > > Thanks for your reply. If I don't include the spin-orbital coupling term > and > > just want to see how are the spin up /down components of the wavefunction > > distributed on the graphene surface, is it possible to do it by > > 1. change the n.n. hopping from -t to -t*sigma0 > > 2. just plot out the even/odd index of the wavefunction (just like in > > > https://mailman-mail5.webfaction.com/pipermail/kwant-discuss/2013-September… > )? > > > > -Originally, I have 86 wavefunctions in total (since there are 86 > > eigenvalues). > > -By using the above approach, I get 172 wavefunctions (now there are 172 > > eigenvalues, ). > > -But when I sum the ground state wavefunction together, I cannot recover > the > > original ground state wavefunction. > > > > What should be correct way to do it? Any hint for the mistakes is > > appreciated. Thanks. > > > > Best, > > Johnny > > > > On 22 November 2015 at 20:18, Anton Akhmerov <anton.akhmerov(a)gmail.com> > > wrote: > >> > >> Dear Johnny, > >> > >> Yes, you can superimpose more plots on top of each other, by providing > >> axes returned by one plot as an argument to the next one. If you want > >> to do something with spin you probably should include spin-orbit > >> coupling, yes. > >> > >> Best, > >> Anton > >> > >> On Sun, Nov 22, 2015 at 10:22 AM, T.C. Wu <tcwu(a)connect.ust.hk> wrote: > >> > Dear Kwant developers and users, > >> > > >> > For the plot of the graphene wavefunction as illustrated in > >> > http://kwant-project.org/doc/1.0/tutorial/tutorial6, is it possible > to > >> > show > >> > the distribution of the spin in the same plot? Should I include the > spin > >> > orbital coupling? Thanks. > >> > > >> > Best, > >> > Johnny > > > > >

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Hopping parameters for multilayered system
by Sergey Nikolaev 26 Nov '15

26 Nov '15

Dear all, Could you please explain me how I can put hopping parameters for a multilayered system with kwant.builder.HoppingKind(). The difficulty is that the intralayer hopping parameters are different for each layer and depend on the layer's position z. T he hopping integrals between neighbouring layers are the same. Thank you in advance. Cheers, Sergey

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Non-interrupted leads
by Oscar Erlandsson 26 Nov '15

26 Nov '15

Hello all, I'm trying to better understand the requirement that 'Builder should interrupt the lead(s)'. I am not fully sure whether my question is a KWANT question or a physics question. Let's say I want to model a 2D system with 'wide leads' like illustrated in the attached figure, i.e. with leads wider than the scattering region, and use a code like the one provided in the 'tutorial/quantum_wire.py' example. If I use the code for j in xrange(W+1): left_lead[lat(0, j)] = 4 * t and so on, where I have replaced 'W' with 'W+1' then attach_lead gives the error "ValueError: Builder does not interrupt the lead, this lead cannot be attached.". If I try another approach where I initially increase the width of the scattering region, attach the leads with the same width as the scattering region, and then use for i in xrange(L): del sys[lat(i, W-1)] del sys[lat(i, 0)] to "peel off" the scattering region edges, sys.finalized() gives me the error "KeyError: Site(kwant.lattice.Monatomic([[1.0, 0.0], [0.0, 1.0]], [0.0, 0.0], ''), array([0, 9]))". I also want to note that I have read a previous thread called "3D structure construction with leads" from 2014 where related issues are discussed. In that thread, Anton Akhmerov writes that "if I follow the lead hoppings from unit cells that are far from the system (say -infinity), then eventually every possible path should end up at a system site, and there should be no path to +infinity that avoids the system" for the leads to be attached meaningfully. I believe my situation satisfies this condition. Is there a reason that it does not make sense to model a system the way I am trying to? If not, how do I go about doing it in KWANT? Thanks, Oscar Erlandsson

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Distribution of spin in graphene lattice
by T.C. Wu 22 Nov '15

22 Nov '15

Dear Kwant developers and users, For the plot of the graphene wavefunction as illustrated in http://kwant-project.org/doc/1.0/tutorial/tutorial6, is it possible to show the distribution of the spin in the same plot? Should I include the spin orbital coupling? Thanks. Best, Johnny

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