Mailman 3 November 2017 - Kwant-discuss

Gate-defined Nanowire Quantum Dot
by Jonathan Fields 15 Sep '22

15 Sep '22

Dear Reader, I am new to KWANT and trying to figure out if it is suitable for calculating transport properties of gate-defined quantum dots made out of nanowires, along the lines of the structure used in https://arxiv.org/abs/cond-mat/0609463. Now, I understand that the mailing list is not a place to ask people to do my work for me, and my question is also not for someone to do this. What I am looking for is some insights into if this is possible (and not all that difficult) with the package. At first sight (and going through the tutorials as well as the APS March Meeting material) this does not seem straightforward; what I struggle with is how to define this type of dot in KWANT. Now, one way would of course be to built the entire 3D geometry of the system, but this will be computationally expensive, and probably also rather tricky to do in terms of defining everything, from the hexagonal wire itself to all of the gates and oxide layers and such. Some abstraction would be preferred, perhaps even reducing the dimensionality and making a 2D system, such as often done in the tutorial. Is this a good idea? My problem is that if one does this, then I run into the problem of how to model the 'half wrap-around' type gates typically used in these devices. In the transport through barrier section of the APS March Meeting material there is a section on how to model realistic potentials, but this would not work all that well for these wrap around type gates. On the other hand, in a way they are perhaps not so different from the QPC in that section. One could model the three gates as something like https://i.imgur.com/WZC983c.png as they do essentially form channels in the wire. Apart from if this sacrifices too much of the nanowire nature in favor of a 2DEG system, I do suppose such a system should in principle be able to be treated as a quantum dot. I haven't been able to confirm this as I am not yet sure how one can apply a source-drain voltage in KWANT; I first thought that this would probably be related to the energy of the modes, but then again I have not been able to produce Coulomb diamonds in this way. The question is getting rather lenghty, and also a bit unclear at this point. Perhaps I should finish up by stating it in a concise form; would you think that it is possible to simulate the transport of such a gate defined nanowire quantum dot device with KWANT, and if so, is the approach I am suggesting above a viable one, or would you go about it very differently? Kind regards Jonathan <http://aka.ms/weboutlook>

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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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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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making systems with translational symmetry perpendicular to the direction of transport
by Maurer, Leon 12 Dec '17

12 Dec '17

Hi all, This is a follow up to my post yesterday (where I put a Si tight binding model into Kwant). My goal is to create a 3D system with translational symmetries in a plane so that I can look at transport perpendicular to that (infinite) plane. So, there wouldn’t be a translational symmetry in the direction of transport, but there will be translational symmetry in the two dimensions perpendicular to transport. It’s a little difficult to explain how I’m trying to do this, but hopefully the below explanation mostly conveys the idea. The attached jupyter notebook uses a standard Si primitive cell as the unit cell, and for simplicity, I’ve chosen the axis of the system to be along the first Si primitive vector as defined in the code (1/2, 1/2, 0). There is translational symmetry along the other two primitive vectors (set in “In [5]” of the notebook). Everything seems to work basically as expected. The axis of the system does lie along the first Si primitive vector; I can attach leads with the same symmetries; and the leads have the right band structure: the band structure down the axis of the leads is the band structure between the gamma and L points (calculated in “In [8]”– compare with the full band structure in “In [10]”). I could calculate the transmission down the axis of the structure with something like kwant.smatrix(syst, energy, args=(0,0)).transmission(1, 0), which isn’t shown in the notebook. (I’d also be putting different potentials in the different sites so that the structure isn’t uniform. There’s not much point calculating transport thru the system in the notebook.) Two questions: 1. Can I calculate transport in other directions --- such as the (1,0,0) direction --- using this system? 2. The primitive lattice vectors are not orthogonal, so I don’t really have translational symmetry perpendicular to the axis of the system. Does anyone have an idea for how to fix that? I could create a (larger) non-primitive unit cell with orthogonal primitive vectors by having a bunch of basis vectors (that sounds weird, but I think that the terminology is right), but I like having my unit cell be simple. (I have tried effectively building a larger unit cell from the primitive cells, but I’ve run into problems – a topic for a different email.) -Leon

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Zero energy states with KPM
by Antonio Lucas Rigotti Manesco 28 Nov '17

28 Nov '17

Dear Kwant users, I am trying to obtain the local density of states of a topological (finite) system by using KPM approximation. My problem is that the gap is too small (as I am considering realistic parameters) and this results in more states than the topologically protected ones appearing for zero energy LDoS. If I control the energy resolution the memory consumption increases too much. So my question is if there is any parameter I can control to have a better energy resolution only near zero energy or maybe somewhere I can compensate the memory consumption (for example, losing resolution at the local distribution in order to obtain better energy resolution). Thanks in advance. -- Antônio Lucas Rigotti Manesco PhD fellow - University of São Paulo, Brazil

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<no subject>
by Prof. CHAN Kwok Sum 24 Nov '17

24 Nov '17

Dear Kwant developer, May I have some references for the method used to find the scattering wavefunction? In the paper in New Journal of Physics about Kwant, a paper about the theory is mentioned. However, I cannot find the paper. Is it possible to have some references on the theory? Thank you in advance. KS Chan Disclaimer: This email (including any attachments) is for the use of the intended recipient only and may contain confidential information and/or copyright material. If you are not the intended recipient, please notify the sender immediately and delete this email and all copies from your system. Any unauthorized use, disclosure, reproduction, copying, distribution, or other form of unauthorized dissemination of the contents is expressly prohibited.

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interpolate between different gauge choices
by Tibor Sekera 24 Nov '17

24 Nov '17

Dear all, I want to set up a 2D system with an out-of-plane magnetic field, where two leads that are perpendicular to each other are attached to the scattering region ('L' shaped system). Because one of the leads is translationally invariant along x direction, while the other one along y direction, I need to interpolate between two different gauge choices. Is there an easy (universal) way to do this? Thanks for any hint, Tibor

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electrical conductivity
by Chung Tsai 24 Nov '17

24 Nov '17

Dear all, I am trying to calculate the electrical conductivity in a cubic system in different temperatures, where chemical potential is set to zero (see below). I am using some of the tips given in the mailing list, I have a couple of questions though. 1- When I solve V=RI to get the voltages, the voltage values are very large. Does it make sense or am I making a mistake somewhere? 2- In order to calculate the conductance, I need to integrate over an energy interval. should I integrate over the entire energy spectrum of the system? The code is very slow, how do I choose the energy interval and the step (least requirements) to ensure something physically meaningful yet speed up the code, any idea? Bests Chung -------------------------------------------------------- import matplotlib.pyplot as pyplot from scipy.spatial import * from matplotlib import rcParams from numpy import * from numpy.linalg import * import pickle import sys import os import string import heapq import kwant import time import itertools as IT from scipy.spatial import cKDTree start = time.time() sigma_x = array([[0, 1], [1, 0]]) sigma_y = array([[0, -1j], [1j, 0]]) sigma_0 = array([[1, 0], [0, 1]]) sigma_z = array([[1, 0], [0, -1]]) a=1.0 t=1.0 L = 4 cutoff_dist = 8 theta=0.24434609527920614 lam=1.0E-3 v0=0.1 hbns= 6.582119514E-16*9.0E22 alpha = 4. mu = 0. e0 = 0.6 nu=3.0E11 K=8.6173303E-5 nuoft = 1 #tmp = 40. se = zeros((int(L**3.), nuoft), float) mo = zeros((int(L**3.), nuoft), float) for ii in range(nuoft): se[:,ii] = linspace(-e0, e0, int(L**3.)) random.shuffle(se[:,ii]) mo[:,ii] = random.uniform(0., theta, int(L**3.)) TL=arange(20, 322, 20) pltl = TL**(-0.25) sigma_xx=[] sigma_xy=[] tosim = 3.8740458655E-5 fintemp=True if fintemp: def dF(energy, Ef, beta): return 0.25*beta/(cosh(0.5*beta*(energy-Ef))**2) def get_trans(beta, lin, lout, Ef=0.): energies=linspace(-e0, e0, 10) #energies=[i/(15*beta) for i in range (-150,150)] trans=[] for energy in energies: smatrix = kwant.smatrix(sys, energy) T=smatrix.transmission(lin, lout) trans.append(T) return trans,energies def Use_temp(beta, lin, lout, Ef=0.): trans,energies=get_trans(beta, lin, lout, Ef=0.) spd=abs(energies[0]-energies[1]) couple_T_E=list(zip(trans,energies)) Conductance=[] Tr=[T*dF(energy,Ef=Ef,beta=beta) for T,energy in couple_T_E] g=trapz(Tr, energies, dx=spd) return g def clcte_sigmas(sys): G = [[Use_temp(beta, i, j, Ef=0.) for i in range(6)] for j in range(6)] G -= diag(sum(G, axis=0)) temp = [0, 1, 2, 3, 4] G = G[temp, :] G = G[:, temp] r = linalg.inv(G) V = r @ array([1, -1, 0, 0, 0]) print(V) E_y = V[3] - V[2] E_x = V[1] - V[0] s_xx = (E_x / (E_x**2 + E_y**2))#*(tosim/(L*1E-7)) s_xy = (E_y / (E_x**2 + E_y**2))#*(tosim/(L*1E-7)) sigma_xx.append(s_xx) sigma_xy.append(s_xy) return s_xx, s_xy def spin_conductance(G, lead_out, lead_in, sigma): """Calculate the spin conductance between two leads. Parameters ---------- G : an instance of `kwant.solvers.common.GreensFunction` The Greens function of the system as returned by `kwant.greens_function`. lead_out : integer The lead where spin current is collected lead_in : integer The lead where spin current is injected sigma : `numpy.ndarray` of shape (2, 2) The Pauli matrix of the quantization axis along which to measure the spin current Notes ----- Calculates the spin conductance between two leads p and q according to: G_{pq} = Tr[ σ_{α} Γ_{q} G_{qp} Γ_{p} G^†_{qp} ] Where Γ_{q} is the coupling matrix to lead q ( = i[Σ - Σ^†] ) and G_{qp} is the submatrix of the system's Greens function between sites interfacing with lead p and q. (see also the function below (spin_conductance2))""" ttdagger = G._a_ttdagger_a_inv(lead_out, lead_in) sigma_z_matrix = kron(eye(ttdagger.shape[0]//2), sigma) return trace(sigma_z_matrix.dot(ttdagger)).real def family_color(sites): return 'black' def hopping_lw(site1, site2): return 0.08 def onsite(site): oe=2.0*e0* kwant.digest.uniform(repr(site)) if oe <= e0: return oe*sigma_0 else: return (e0-oe)*sigma_0 def hop1(site1, site2, tempt): ni = int(site1[0]*L**2. + site1[1]*L + site1[2]) nj = int(site2[0]*L**2. + site2[1]*L + site2[2]) dij = norm(array(site1 - site2)) rs = dij**2.0 eij = (abs(se[ni,0]-mu)+abs(se[nj,0]-mu)+abs(se[ni,0]-se[nj,0])) zij = nu*exp(-2.0*alpha*(dij))*exp(-(0.5/(K*tempt))*eij) return zij*sigma_0 def cuboid_shape(pos): x, y, z = pos return 0 <= x < L and 0 <= y < L and 0 <= z < L def xlead_shape(pos): x, y, z = pos return x==0 and 0 <= y < L and 0 <= z < L def ylead1_shape(pos): x, y, z = pos return 0 <= x < L/2 and y==0 and 0 <= z < L def ylead2_shape(pos): x, y, z = pos return round(L/2) <= x < L and y==0 and 0 <= z < L prim_vecs=array([(a,0.,0.),(0.,a,0.),(0.,0.,a)]) offset1=array((0.0, 0.0, 0.0)) lat=kwant.lattice.Monatomic(prim_vecs, offset1, norbs=2) for tmp in TL: print('T=',tmp,'K') print('The conductivity is being calculated') sys = kwant.Builder() beta = 1.0/(K*tmp) sys[lat.shape(cuboid_shape, (0, 0, 0))] = 0*sigma_0 sites = set(sys.sites()) for i in sites: sys[i] = se[int(i.tag[0]*L**2. + i.tag[1]*L + i.tag[2]),0]*sigma_0 for jump in IT.combinations(sites, 2): sys[jump[0], jump[1]] = hop1(jump[0].tag, jump[1].tag, tmp) lead0 = kwant.Builder(kwant.TranslationalSymmetry((-a, 0, 0))) lead0[lat.shape(xlead_shape, (0, 0, 0))] = 0*sigma_0 lead0[lat.neighbors()] = -t*sigma_0 sys.attach_lead(lead0) sys.attach_lead(lead0.reversed()) lead2 = kwant.Builder(kwant.TranslationalSymmetry((0, -a, 0))) lead2[lat.shape(ylead1_shape, (0, 0, 0))] = 0*sigma_0 lead2[lat.neighbors()] = -t*sigma_0 sys.attach_lead(lead2) sys.attach_lead(lead2.reversed()) lead3 = kwant.Builder(kwant.TranslationalSymmetry((0, -a, 0))) lead3[lat.shape(ylead2_shape, (L-1, 0, 0))] = 0*sigma_0 lead3[lat.neighbors()] = -t*sigma_0 sys.attach_lead(lead3) sys.attach_lead(lead3.reversed()) system=kwant.plot(sys, site_lw=0.0001, site_color=family_color, hop_lw=hopping_lw) sys = sys.finalized() s_xx, s_xy = clcte_sigmas(sys) print('σ_xx=', s_xx, 'σ_xy=', s_xy) print() end = time.time() print() print ("The execution time:") if (end - start < 60): print (end - start, "seconds") if (end - start > 60) and (end - start < 3600): print ((end - start)/60, "minutes") if (end - start > 3600) : print ((end - start)/3600, "hours") print()

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confusion about up/down conductance in the new version of kwant
by SUDIN GANGULY 22 Nov '17

22 Nov '17

Dear Sir, I was trying to compute the transmissions for a 2 terminal GNR from lead 0 (up/down) to lead 1 (up/down) using the following discussion. http://kwant-discuss.kwant-project.narkive.com/ZRoAKlBg/up-down-conductance I constructed the leads as follows with the help of tutorial of superconductor: def create_lead_h(W, symmetry, axis=(0, 0)): lead = kwant.Builder(symmetry,conservation_law=-sigma_z) lead[lat.wire(axis, W)] = 0. * sigma_z lead[lat.neighbors(1)] = -1.*sigma_z return lead and have calculated the following transmissions: smatrix = kwant.smatrix(sys, e, args=(params,)) su = smatrix.transmission((1, 0), (0, 0)) + smatrix.transmission((1, 0), (0, 1)) sd = smatrix.transmission((1, 1), (0, 0)) + smatrix.transmission((1, 1), (0, 1)) cc = smatrix.transmission(1, 0) The system has Rashba interaction. My concern is when I turn off the Rashba interaction, su and sd should be equal to each other and have integer values. But I'm getting the integer values only for su. I don't understand why I'm getting this behaviour. The program is written here for better understanding. ================================================================= from math import sqrt import random import itertools as it import tinyarray as ta import numpy as np from matplotlib import pyplot import kwant #from sympy import S, Eq, solve, Symbol class Honeycomb(kwant.lattice.Polyatomic): """Honeycomb lattice with methods for dealing with hexagons""" def __init__(self, name=''): prim_vecs = [[0.5, sqrt(3)/2], [1, 0]] # bravais lattice vectors # offset the lattice so that it is symmetric around x and y axes basis_vecs = [[-0.5, -1/sqrt(12)], [-0.5, 1/sqrt(12)]] super(Honeycomb, self).__init__(prim_vecs, basis_vecs, name, norbs=2) self.a, self.b = self.sublattices def hexagon(self, tag): """ Get sites belonging to hexagon with the given tag. Returns sites in counter-clockwise order starting from the lower-left site. """ tag = ta.array(tag) # a-sites b-sites deltas = [(0, 0), (0, 0), (1, 0), (0, 1), (0, 1), (-1, 1)] lats = it.cycle(self.sublattices) return (lat(*(tag + delta)) for lat, delta in zip(lats, deltas)) def hexagon_neighbors(self, tag, n=1): """ Get n'th nearest neighbor hoppings within the hexagon with the given tag. """ hex_sites = list(self.hexagon(tag)) return ((hex_sites[(i+n)%6], hex_sites[i%6]) for i in range(6)) def cross(W, L): def shape(pos): return ((-W <= pos[1] <= W and -L <= pos[0] <= L)) # horizontal return shape def circle(W,L,r1,r2): def ring(pos): (x, y) = pos rsq = x ** 2 + y ** 2 if (sqrt(rsq)>=r1): return (-W <= pos[1] <= W and -L <= pos[0] <= L) return ring ## Pauli matrices ## sigma_0 = np.array([[1, 0], [0, 1]]) # identity sigma_x = np.array([[0, 1], [1, 0]]) sigma_y = np.array([[0, -1j], [1j, 0]]) sigma_z = np.array([[1, 0], [0, -1]]) ## Hamiltonian value functions ## def onsite_potential(site, params): return params['ep'] * sigma_0 def potential_shift(site, params): return -params['mu'] * sigma_0 def kinetic(site_i, site_j, params): return -params['gamma'] * sigma_0 def rashba(site_i, site_j, params): d_ij = site_j.pos - site_i.pos rashba = 1j * params['V_R'] * (sigma_x * d_ij[1] - sigma_y * d_ij[0]) return rashba + kinetic(site_i, site_j, params) def spin_orbit(site_i, site_j, params): so = 1j * params['V_I'] * sigma_z return so lat = Honeycomb() pv1, pv2 = lat.prim_vecs ysym1 = kwant.TranslationalSymmetry(pv2 - 2*pv1) # lattice symmetry in -y direction ysym2 = kwant.TranslationalSymmetry(-pv2 + 2*pv1) # lattice symmetry in -y direction xsym1 = kwant.TranslationalSymmetry((-pv2)) # lattice symmetry in -x direction xsym1.add_site_family(lat.sublattices[0], other_vectors=[(-2, 1)]) xsym1.add_site_family(lat.sublattices[1], other_vectors=[(-2, 1)]) xsym2=kwant.TranslationalSymmetry((pv2)) xsym2.add_site_family(lat.sublattices[0], other_vectors=[(-2, 1)]) xsym2.add_site_family(lat.sublattices[1], other_vectors=[(-2, 1)]) def create_lead_h(W, symmetry, axis=(0, 0)): lead = kwant.Builder(symmetry,conservation_law=-sigma_z) lead[lat.wire(axis, W)] = 0. * sigma_z lead[lat.neighbors(1)] = -1.*sigma_z#kinetic return lead def create_lead_v(L,W, symmetry): axis=(0, 0) lead = kwant.Builder(symmetry,conservation_law=-sigma_z) lead[lat.wire(axis, W)] = 0. * sigma_0 lead[lat.neighbors(1)] = kinetic return lead def create_hall_cross(W, L): ## scattering region ## sys = kwant.Builder() sys[lat.shape(circle(W,L,r1,r2), (0, r1+1))] = onsite_potential sys[lat.neighbors(1)] = rashba sys[lat.neighbors(2)] = spin_orbit ## leads ## leads = [create_lead_h(W, xsym1)] leads += [create_lead_h(W, xsym2 for lead in leads: sys.attach_lead(lead) return sys #========================================================================= def spin_conductance(G, lead_out, lead_in, sigma): """Calculate the spin conductance between two leads. Parameters ---------- G : an instance of `kwant.solvers.common.GreensFunction` The Greens function of the system as returned by `kwant.greens_function`. lead_out : integer The lead where spin current is collected lead_in : integer The lead where spin current is injected sigma : `numpy.ndarray` of shape (2, 2) The Pauli matrix of the quantization axis along which to measure the spin current Notes ----- Calculates the spin conductance between two leads p and q according to: G_{pq} = Tr[ σ_{α} Γ_{q} G_{qp} Γ_{p} G^+_{qp} ] Where Γ_{q} is the coupling matrix to lead q ( = i[Σ - Σ^+] ) and G_{qp} is the submatrix of the system's Greens function between sites interfacing with lead p and q. """ ttdagger = G._a_ttdagger_a_inv(lead_out, lead_in) sigma_z_matrix = np.kron(np.eye(ttdagger.shape[0]//2), sigma) return np.trace(sigma_z_matrix.dot(ttdagger)).real#, np.trace(ttdagger).real if __name__ == '__main__': params = dict(gamma=1., ep=0, mu=0.0, V_I=0.0, V_R=0.0) W=17.5 L=20 r1=0 r2=10 sys = create_hall_cross(W, L) kwant.plot(sys, site_color=site_color, site_size=site_size) sys=sys.finalized() band=100 Es = np.linspace(-0.5, 0.5, band) sp_up=[] sp_down=[] ccurrent=[] spz=[] for e in Es: G = kwant.greens_function(sys, e, args=(params,)) smatrix = kwant.smatrix(sys, e, args=(params,)) su = smatrix.transmission((1, 0), (0, 0)) + smatrix.transmission((1, 0), (0, 1)) sd = smatrix.transmission((1, 1), (0, 0)) + smatrix.transmission((1, 1), (0, 1)) cc = smatrix.transmission(1, 0) spin_pol_z = spin_conductance(G, 1, 0, sigma_z) diff=su-sd print (e,su,sd,cc,spin_pol_z,diff) sp_up.append(su) sp_down.append(sd) ccurrent.append(cc) spz.append(spin_pol_z) pyplot.figure() pyplot.plot(Es, sp_up) pyplot.plot(Es, sp_down) pyplot.plot(Es, ccurrent) pyplot.xlabel("energy [t]") pyplot.ylabel("spin polarization_x [e^2/h]") pyplot.show() ============================================================= With Regards, Sudin

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