Kwant-discuss February 2022

kwant-discuss@python.org

7 participants
12 discussions

current-voltage diagram
by loojoo026＠gmail.com 08 Apr '22

08 Apr '22

dear all I have done my project using Kwant and I have drawn a conductance-energy diagram for the system. The question I have is how can I get the current-voltage diagram? thanks Leo

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Inquiry about extracting boundstate eigenvectors/eigenvalues from kwant
by Zhang, Chi 17 Mar '22

17 Mar '22

Dear Devs, I hope this email finds you well. We are two Master's students trying to calculate Andreev/Majorana transition elements using the kwant boundstate algorithm (doi: 10.21468/SciPostPhys.4.5.026). For our purposes we need to extract the evanescent tails of the boundstates in the leads as well, which we had hoped to obtain from the StabilisedModes object in kwant.InfiniteSystem.modes(). However, in general, the vecs and vecslmbdainv object are not related by a constant eigenvalue, as advertised in the documentation. We wonder if you could tell us how to transform from the vecs and vecslmbdainv objects back to the eigenvectors*eigenvalues and eigenvectors object for our calculations? Many thanks in advance for your response! Sincerely, Chi Zhang & Ryan Tiew MSci Physics with Theoretical Physics Imperial College London ----------------------------------------------------- Some technical details of what we have tried here. Please ignore this if you don't have enough time to read. We are aware that kwant is transforming vecslmbdainv by sqrt(norm(hop))*eigenvectors, and vecs by (hop/sqrt(norm(hop)))@eigenvectors@eigenvalues. However, this does not seem to be the full story and we are not able to reproduce what is stored in vecs and vecslmbdainv when there are more than 1 evanescent modes. In particular, we have tried to do a real generalised Schur decomposition of the two matrices in the lead schroedinger's equation, grabbed the vectors stored in the right unitary, and put them through the basis transformation described above. Still the there is no sign of better agreements. We therefore we believe it would be more productive to inquire directly from you how to transform from vecs and vecslmbdainv to actual eigenvectors and eigenvalues. Many thanks in advance for your reply.

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How the spin is positioned in the Smatrix?
by Qingtian Zhang 06 Mar '22

06 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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Spin polarized transport calculation
by L.M J 06 Mar '22

06 Mar '22

Dear Kwant users and developers I have two questions related to the spin calculation using kwant. 1. I'm wondering if we can do spin-polarized transport in kwant? For example, can we can get the separate Conductance Vs Energy diagram for spin up and spin down terms. I'm thinking one way of doing this is to set up the Rashba Hamiltonian as a whole. And then manually extract the spin up and spin down hamiltonian and do the transport calculation separately. But this will eliminate some extra terms that I don't know whether this is correct anymore. Any suggestions? 2. How can we set up the spin polarization for a particular site? For example, in some of the topological insulator, can we set up spin polarization on the edge as up and the other side of the edge as down, and then do the transport? Or this question is completely wrong? Thanks for advance. Sincerely, Liming

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Parallel Computation
by diegobruno244＠gmail.com 03 Mar '22

03 Mar '22

Kwant Community, I would like to know if it is possible to do parallel computing using kwant? To do parallel computing can be used numba, CUDA, multiprocessing, Parallel. This work for kwant? And if possible are there any examples?

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Andreev Reflection in Graphene
by joiss＠sunypoly.edu 23 Feb '22

23 Feb '22

Hello All, I have seen that this problem has been discussed in threads before and thanks to them, I've managed to set up most of it. But I need help on one main issue I'm seeing right now. What is unusual right now is by setting any non-zero fermi-energy in the normal metal lead, the bands split and a gap opens. I wonder if I am using the wrong matrices. Code below: import kwant import numpy as np from matplotlib import pyplot as plt from math import pi, sqrt, tanh import tinyarray % matplotlib notebook ####___Fundemental Constants___#### e = 1.60217662 *(10**-19) # hbar = 1.0545718*(10**-34) # h = 6.62607004*(10**-34) # e_hb = e/hbar # ################################### #Specifying 2x2 Pauli matrices acting on spin s_x = tinyarray.array([[0,1],[1,0]]) s_y = tinyarray.array([[0,-1j],[1j,0]]) s_z = tinyarray.array([[1,0],[0,-1]]) s_0 = tinyarray.array([[1,0],[0,1]]) #Specifying 2x2 Pauli matrices acting on particle-hole tau_0 = tinyarray.array([[1, 0], [0, 1]]) tau_x = tinyarray.array([[0, 1], [1, 0]]) tau_y = tinyarray.array([[0, -1j], [1j, 0]]) tau_z = tinyarray.array([[1, 0], [0, -1]]) delta_gap = 0.025 class SimpleNamespace(object): def __init__(self, **kwargs): self.__dict__.update(kwargs) # Ef_S is the fermi level in the superconducting region # Ef is the fermi level in the normal graphene region def make_system(a=0.142, W=1, L=1, t=2.71, tp=-0.1, V0=0.001, Ef=delta_gap*1, Ef_S=delta_gap*10, Delta=delta_gap): delta_t = lambda theta: Delta * np.cos(theta) lat = kwant.lattice.general([(a * 1, 0), (a * 0.5, a * 0.5 * np.sqrt(3))], [(0, 0), (0, a * 1 / np.sqrt(3))], norbs=4) a1, b1 = lat.sublattices def channel(pos): x, y = pos return -W <= y <= W and -L <= x <= L syst = kwant.Builder() syst[lat.shape(channel, (0, 0))] = Ef * -1 * np.kron(tau_z,s_0) syst[lat.neighbors()] = t *-1* np.kron(tau_z,s_0) syst.eradicate_dangling() def lead0_shape(pos): # top lead x,y= pos return -L <= x <= L sym_up = kwant.TranslationalSymmetry(lat.vec((-1, 2))) # Normal graphene lead-0 lead0 = kwant.Builder(sym_up, conservation_law= np.kron(tau_z,s_0), particle_hole=np.kron(tau_x,s_0)) lead0[lat.shape(lead0_shape, (0, 0))] = Ef *-1* np.kron(tau_z,s_0) lead0[lat.neighbors()] = t *-1* np.kron(tau_z,s_0) def lead1_shape(pos): # bottom lead x,y = pos return -L <= x <= L sym_down = kwant.TranslationalSymmetry(lat.vec((1, -2))) # bottom lead # Superconducting graphene lead-1 lead1 = kwant.Builder(sym_down) lead1[lat.shape(lead1_shape, (0, 0))] = (Ef_S + V0) *-1* np.kron(tau_z,s_0) + Delta * -1 * np.kron(tau_y,s_y) lead1[lat.neighbors()] = t *-1* np.kron(tau_z,s_0) return syst, [lead0,lead1] def plot_bandstructure(flead, momenta, header): bands = kwant.physics.Bands(flead) energies = [bands(k) for k in momenta] plt.figure(figsize=(9,6)) plt.plot(momenta, energies) plt.xlabel("k $(1/a_0)$",fontsize=15) plt.xlim(left=-1,right=1) plt.ylabel("E (t)",fontsize=15) plt.ylim(top=1,bottom=-1) plt.tick_params(labelsize=12) plt.title(header) plt.show() def plot_conductance(syst, energies): # Compute transmission as a function of energy data = [] for energy in energies: smatrix = kwant.smatrix(syst, energy) # Conductance is N - R_ee + R_he data.append(smatrix.submatrix((0, 0), (0, 0)).shape[0] - smatrix.transmission((0, 0), (0, 0)) + smatrix.transmission((0, 1), (0, 0))) plt.figure(figsize=(8,6)) plt.plot(energies/delta_gap, data) plt.xlabel("$E/\Delta$",fontsize=15) plt.ylabel("G ($e^2/h$)",fontsize=15) # plt.ylim(bottom=0,top=2.5) plt.tick_params(labelsize=15) plt.show() def main(): a=0.142 E = 0.001 system, leads = make_system() # plot_system(system,a) # Attach the leads to the system. for lead in leads: system.attach_lead(lead) # Plot system with leads def family_colors(site): return 0 if site.family == a else 1 # kwant.plot(system, site_color=family_colors, site_lw=0.1, lead_site_lw=0, colorbar=False) # Finalize the system. system = system.finalized() # Compute the band structure of lead 0. momenta = [-pi + 0.02 * pi * i for i in range(101)] plot_bandstructure(system.leads[0], momenta,'Normal Lead') plot_bandstructure(system.leads[1], momenta,'Superconducting Lead') # Plot conductance. energies = np.linspace(-0,2*delta_gap,201) plot_conductance(system, energies) if __name__ == '__main__': main() Any guidance or corrections will be appreciated! Thank you, Sharadh

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Graphene Conductor-Superconductor Conductance above Gap
by Henry Axt 21 Feb '22

21 Feb '22

Hello, I set up a graphene lattice with periodic boundary conditions that is connected to a superconductor (simulated with an electron and hole lattice). When observing the transmission between the electron and hole lattice, I see a small amount conductance above the superconducting gap. I do not see this with something like a square lattice. What could explain this?

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Units and volume for conductivity using KPM
by Bruno Focassio 20 Feb '22

20 Feb '22

Hello everyone, I'm having difficulty figuring out the correct units conversion factor for the conductivity obtained from the KPM method. I tried following the example for the Haldane model as much as possible. In the link, there is a minimal working example for computing the longitudinal conductivity for platinum https://github.com/bfocassio/ExampleKwantConductivity. I'm using a model derived from Wannier functions for bulk Pt in the conventional unit cell. The question is: Should I normalize the conductivity by the volume per site? (using the volume in m^3?) Are there any other prefactors that I am missing? The goal is to obtain conductivity in SI units.

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A problem in calculating conductivity using the KPM method
by 220202007＠seu.edu.cn 18 Feb '22

18 Feb '22

Dear all： When I use the KPM method in KWANT, the longitudinal conductivity at zero point of graphite is around 1.03, but according to the literature it should be (4/ π).，The difference between the two is large, so I would like to ask what the specific reason is. Thank you for your answer. Best wishes Hanlin Liu

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Surprising (?) behaviour of attach leads
by maciej.zwierzycki＠ifmpan.poznan.pl 15 Feb '22

15 Feb '22

Dear Kwant Authors and Users, Greetings and thank you for a wonderful tool. While trying to force Kwant to accept the unit cell of the lead in a particular, connected form, I've encountered somewhat puzzling situation illustrated in the attached piece of code. The described procedure does not make sense physically, but its purpose is to illustrate the behaviour which may possibly (?) lead to problems, at least for newbies like me. ;) Let's say I define two honeycomb lattices, differing in the choice of basis, but entirely equivalent (i.e. they overlap) prime=np.array([[sqrt(3)/2,sqrt(3)],[1.5,0.0]])*a basis_at=np.array([[0.0,0.0],[-0.5,0.5]])*a lat = kwant.lattice.general([prime[:,0],prime[:,1]],[basis_at[:,0],basis_at[:,1]]) and basis2=np.array([[0.0,sqrt(3)/2],[-0.5,-1.0]])*a lat2 = kwant.lattice.general([prime[:,0],prime[:,1]],[basis2[:,0],basis2[:,1]]) I'm using numpy so that vector algebra works. The rectangular scattering region and the left lead (same width) are populated with "lat". Now I'm adding a few atoms from lat2 to the right edge of the scattering region, e.g. a single hexagon for i in range (0,1): # changing the range we can generate also the full layer of lat2 sites syst[lat2.sublattices[0](0+2*i,9-i)]=0.0 syst[lat2.sublattices[1](0+2*i,8-i)]=0.0 syst[lat2.sublattices[0](1+2*i,8-i)]=0.0 syst[lat2.sublattices[1](1+2*i,8-i)]=0.0 and then attach the lead, also populated with "lat2". I know, that's not how it's described in tutorial. ;) The filling algorithm generates the system which to naked eye looks like the perfect nanowire. The transmission however is not perfect i.e. T < N (number of modes). If the full layer of lat2 sites is added to the right edge, than everything is as it should be, i.e. perfect transmission. I'm not sure if it's a bug - I'm doing things not in the manual - but I find it strange. A perfect system, however generated, should lead to perfect transmission. What gives? Also I noticed that neighbours() method called for one of the lattices tends to pick up the sites from the other lattice, if they belong to he sublattice generated from (0,-0.5a), i.e. from the common basis. Try commenting out one of the lines 53-54. Is it intentional or not? It's quite likely that the described situation never arises in real-world situations especially if one does the right thing and pads the interface with the full principal layer before attaching the leads. I've stumbled upon it only thanks to my laziness. But it might also illustrate some fragility of the algorithm, possibly worth eliminating. In any case I'd be grateful for some illumination. Why isn't the "perfect" wire behaving as it should? Where is the imperfection if it's not visible in the structure? With Kind Regards Maciej Zwierzycki import kwant import numpy as np from numpy import sin, cos,tan,sqrt from matplotlib import pyplot def make_system(a, t, W, L1): prime=np.array([[sqrt(3)/2,sqrt(3)],[1.5,0.0]])*a basis_at=np.array([[0.0,0.0],[-0.5,0.5]])*a lat = kwant.lattice.general([prime[:,0],prime[:,1]],[basis_at[:,0],basis_at[:,1]]) syst = kwant.Builder() #### Define the scattering region. #### def scatt_shape(pos): (x, y) = pos return (-L1 < x < L1) and ( -W <y < W) syst[lat.shape(scatt_shape,basis_at[:,0])]=0 syst[lat.neighbors()] = t #### Left lead. #### def lead_shape(pos): (x, y) = pos return (-W < y < W ) lsym=kwant.TranslationalSymmetry(lat.vec((0,-1))) lsym.add_site_family(lat.sublattices[0], other_vectors=[(-2,1)]) lsym.add_site_family(lat.sublattices[1], other_vectors=[(-2,1)]) llead = kwant.Builder(lsym) llead[lat.shape(lead_shape,basis_at[:,0])] = 0 llead[lat.neighbors()] = t #### Right lead. #### ### Define a hexagonal lattice with different - but equivalent! - basis basis2=np.array([[0.0,sqrt(3)/2],[-0.5,-1.0]])*a lat2 = kwant.lattice.general([prime[:,0],prime[:,1]],[basis2[:,0],basis2[:,1]]) #### Extra hexagon of lat2 on the right edge of the scattering region: #### -- range(0,1) - single hexagon ##### -- range(-2,3) - full width for i in range (0,1): syst[lat2.sublattices[0](0+2*i,9-i)]=0.0 syst[lat2.sublattices[1](0+2*i,8-i)]=0.0 syst[lat2.sublattices[0](1+2*i,8-i)]=0.0 syst[lat2.sublattices[1](1+2*i,8-i)]=0.0 syst[lat2.neighbors()]=t syst[lat.neighbors()]=t kwant.plot(syst) lsym=kwant.TranslationalSymmetry(lat2.vec((0,1))) lsym.add_site_family(lat2.sublattices[0], other_vectors=[(2,-1)]) lsym.add_site_family(lat2.sublattices[1], other_vectors=[(2,-1)]) rlead= kwant.Builder(lsym) rlead[lat2.shape(lead_shape,basis_at[:,0])] = 0 rlead[lat2.neighbors()] =t syst.attach_lead(llead) syst.attach_lead(rlead) syst=syst.finalized() return syst syst = make_system(a=1.42,t=3.0,W=10.5,L1=20) kwant.plot(syst) data1 = [] data2 = [] params = dict(a=1.42, t=-3.0) energies=[0.2*i+0.01 for i in range(20)] for energy in energies: print('En=',energy) smatrix = kwant.smatrix(syst, energy,params=params) data1.append(smatrix.transmission(1, 0)) data2.append(smatrix.num_propagating(0)) pyplot.figure() pyplot.plot(energies, data1, 'r--', label='T') pyplot.plot(energies, data2,'r>', label='N') legend = pyplot.legend(loc='upper left') pyplot.xlabel("energy") pyplot.ylabel("conductance [e^2/h]") pyplot.show()

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Kwant-discuss February 2022