Mailman 3 March 2014 - Kwant-discuss

How the spin is positioned in the Smatrix?
by Qingtian Zhang March 6, 2022

March 6, 2022

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))

3 3

problem with a ring geometry
by Diego Rainis March 25, 2014

March 25, 2014

Dear all, thanks in advance for the support. As I already had occasion to explain to Xavier, my problem concerns a ring geometry where changing Hamiltonian values on non-existing sites inside the ring (r<r1) actually changes the conductance behavior, while it should not. More precisely, I am building a ring, assigning to the ring sites a certain value given by the/onsite/ function, which however in its definition contains a conditional /if-/instruction involving sites inside the ring. I was expecting that the /if-/condition is then never satisfied and thus no change is made, but the conductance curve changes if I change the values of, say, the chemical potential (on-site energy) inside the ring, where no sites exist. However, If I do the /same/ changes /outside/ the ring (r>r2), no change occurs in the conductance, as it should. Of course one hypothesis is that I am in fact changing also Hamiltonian elements of the ring, but I don't see why. As suggested by Xavier, I will try to check that by using plotter. I attach the piece of code. I put a comment on the line where I write the conditional instruction for mu. If you try to run the code for different values of mu_inside, and/or different physical sub-regions inside the ring, you should see different conductance behavior. I am using python 2.7.5 on a Mac OS X 10.6.8 and the same holds true for a Mac OS X 10.8.5. Thanks a lot in advance for any hint on how to understand this misbehavior. Diego ============================================================================== import kwant import scipy, scipy.io from numpy import * from pylab import * import sys from numpy import mod from math import pi,atan2 import matplotlib as mpl from matplotlib import pyplot import random; import pickle params = {'backend': 'pdf', 'ytick.direction': 'out', 'text.usetex': True, 'font.size':16} rcParams.update(params) matplotlib.rcParams.update({'font.size': 20}) im=0+1j import tinyarray import scipy.sparse.linalg class SimpleNamespace(object): """A simple container for parameters.""" def __init__(self, **kwargs): self.__dict__.update(kwargs) s_0 = identity(2) s_z = array([[1, 0], [0, -1]]) s_x = array([[0, 1], [1, 0]]) s_y = array([[0, -1j], [1j, 0]]) #================================================================================= def make_system(a=1, W=10, r1=10, r2=20): lat = kwant.lattice.square(a) sys = kwant.Builder() def ring(pos): (x, y) = pos rsq = x ** 2 + y ** 2 return (r1 ** 2 < rsq < r2 ** 2) def mu(site, p): (x, y) = site.pos if abs(x) < r1 and abs(y) < r1: return p.mu_inside # this is the critical line: changing mu values inside the ring should not change the ring conductance else: return p.mu_ring def onsite(site, p): return ( 4.0 * p.t - mu(site, p)) * s_0 def hopping(site0, site1, p): return -p.t * s_0 # =============================================================================== sys[lat.shape(ring, (0, r1 + 1))] = onsite sys[lat.neighbors()] = hopping # =================================================================================== sym_lead = kwant.TranslationalSymmetry((-a, 0)) lead = kwant.Builder(sym_lead) def lead_shape(pos): (x, y) = pos return (-W / 2 < y < W / 2) lead[lat.shape(lead_shape, (0, 0))] = onsite lead[lat.neighbors()] = hopping sys.attach_lead(lead) sys.attach_lead(lead.reversed()) return sys def plot_conductance(sys, energies, p): data = [] for energy in energies: smatrix = kwant.smatrix(sys, energy, args=[p]) data.append(smatrix.transmission(1, 0)) axes([0.12, 0.12, 0.8, 0.8]) hold(True) pyplot.plot(energies, data, 'bo-', ms=2.8, linewidth=0.5,markeredgewidth=0.03) pyplot.axis([0, 3.0, 0, 10.1]) pyplot.xlabel(r'$eV[t]$') pyplot.ylabel(r'$G[e^2/h]$') savefig('Ring_G(e).pdf') def main(): sys = make_system() kwant.plot(sys) sys = sys.finalized() params = SimpleNamespace(t=1.0, mu_ring=0.0, mu_inside=-1.62) pyplot.figure() plot_conductance(sys, energies=[0.01 * i for i in xrange(300)], p=params) if __name__ == '__main__': main() ==================================================================================

5 8

Transport with spin
by Abbout Adel March 17, 2014

March 17, 2014

Dear all, I have two simple questions concerning electronic transport with spins with no interaction: If I take the program spin_orbit.py from the tutorial and call the system sys=make_system ( alpha=0, ez=0, W=2, L=2) # no spin orbit interaction print matrix.round(kwant.smatrix(sys,energy=1.7).data , 2 ) I choose the energy E=1.7 because one mode is closed. the result is: [[-0.00-0.j 0.00-0.j 0.41+0.43j -0.74-0.33j] [ 0.00-0.j -0.00-0.j -0.80+0.08j -0.52+0.28j] [ 0.41+0.43j -0.74-0.33j -0.00-0.j 0.00-0.j ] [-0.80+0.08j -0.52+0.28j 0.00-0.j -0.00-0.j ]] What surprises me is that there is scattering between spins knowing that the Hamiltonians for the two spins are separated without interaction ! This seems to appear only when one mode is closed. To understand better and check this result, I rewrote the program using two lattices (as for electrons and holes). the program is attached with the Email. The result for the scattering matrix is [[ 0.00-0.j -0.16+0.99j 0.00+0.j 0.00+0.j ] [-0.16+0.99j 0.00+0.j 0.00+0.j 0.00+0.j ] [ 0.00+0.j 0.00+0.j 0.00+0.j -0.16+0.99j] [ 0.00+0.j 0.00+0.j -0.16+0.99j 0.00+0.j ]] The result is now better because there is no scattering between spins. My Question is :what am I missing? The second question concerns the position of the coefficients in the scattering matrix. We have the Hamiltonian [image: H=h \otimes 1_{2\times2}] where [image: \otimes] is the Kronecker product and [image: h] the spinlessHamiltonian. The scattering matrix is given by : [image: S=-1+2\pi i W^\dagger \frac{1}{E-H-\Sigma} W] [image: W] is the coupling matrix and [image: \Sigma] the self energy If we look at the simple system [image: ( L=2,W=2)] all the matrices are square and we can easily proove, just by the properties of the Kronecker poduct that the scattering matrix can be written as: [image: S=s\otimes 1_{2\times2}] Where [image: s] is the scattering matrix without spin. [image: s= \left(\begin{array} {c c} r & t \\ t^\prime & r^\prime \end{array}\right)] [image: t] is the transmission matrix. So I am expecting a scattering matrix with the followin form: [image: S= \left(\begin{array} {c c c c} r &0 & t & 0 \\ 0 &r & 0 & t \\ t^\prime &0 & r^\prime & 0 \\ 0& t^\prime & 0 & r^\prime \end{array}\right)] This is consistent with the result with two lattices but still the position of the coefficients are different from the one obtained from the Hamiltonian base. So I suggest that you are using different base. The question I ask is: in order to know the position of coefficients corresponding to different modes or different spins, which formula do you use to obtain the scattering matrix. is it a matrix inversion or some recursive method ? Thank you in advance. Regards, A. Abbout

2 3

Re: [Kwant] [PROVENANCE INTERNET] Re: Propagating mode
by Joseph Weston March 13, 2014

March 13, 2014

Hi again, I just realized that my example was kind of broken as I defined a a variable `mode` and then promptly overwrote this with something else. The examples should have read as follows:: lead = 1 # whatever mode = 0 # whatever smatrix = kwant.smatrix(sys, energy) modes = smatrix.lead_info sites = fsys.leads[lead].sites psi = modes[lead].wave_functions[:, mode] psi = dict(zip(sites, psi)) and:: sites = fsys.leads[lead].sites psi = modes[lead].wave_functions[:, mode] psi_up = dict(zip(sites, psi[::2])) # even elements of psi (up) psi_down = dict(zip(sites, psi[1::2])) # odd elements of psi (down) Regards, Joe

1 0

Re: [Kwant] Propagating mode
by christoph.groth＠cea.fr March 13, 2014

March 13, 2014

Hi, An addition to Joseph’s reply: If all you want to do is to save the wave function, you will probably not want to build a dictionary, but simply loop over the wave function array and the list of sites while writing both to a file. For example like this (with spin): sites = fsys.leads[lead].sites psi = modes[lead].wave_functions[:, mode_nr] for site, value in zip(sites, psi.reshape(-1, 2)): tag = site.tag print tag[0], tag[1], value[0], value[1] Note how I used NumPy’s “reshape” to be able to iterate over the spins. Also, bear in mind that the coordinates of the sites that make up a lead are most likely not the ones that you have set, but equivalent ones (under that leads’ symmetry). Best, Christoph

1 0

Propagating mode
by Pawel Wojcik March 13, 2014

March 13, 2014

Dear all, I would like to ask about the wave function of the propagating modes because I don't know how should i write it into the file. I do that using the code smatrix = kwant.smatrix(sys, energy) mode= smatrix.lead_info and now mode[1].wave_functions[i][0] gives as the wave function of the propagating mode corresponding to the 0 orbital in lead 1. I would like to write this wave function into the file but a don't know how is the correspondence between position "i" and the real position in the lead. It is important for me because I need to know how is the average spin of each propagating mode. Thank you in advance Best Regards Pawel

2 1

Current density
by Arash March 6, 2014

March 6, 2014

Hi there, I'm trying to simulate a scattering region confined between two reservoirs(leads) for that purpose I defines a wire with linearly increasing onsite potentials. Now I want to find the local current density at each site on the wire but I don't know how to do it. Can anyone guide me through this? :) def make_system(a=1, t=1.0, W = 60, L = 100, pot=-0.2): lat = kwant.lattice.square(a) sys = kwant.Builder() def hop(site1, site2, B=0): # The magnetic field is controlled by the parameter B x1, y1 = site1.pos x2, y2 = site2.pos return -t * exp(0.5j * B * (y1+y2)* (x1-x2))#exp(0.5j * B * (y1-x1)) def potential_prof(i,L,pot): #potential profile linearly determined here return (L-i)*pot/L # Define the scattering region for i in xrange(L): for j in xrange(W): # On-site Hamiltonian sys[lat(i, j)] = 4 * t+potential_prof(i,L,pot) sys[kwant.builder.HoppingKind((1, 0), lat, lat)] = hop sys[kwant.builder.HoppingKind((0, 1), lat, lat)] = hop sym_left_lead = kwant.TranslationalSymmetry((-a, 0)) left_lead = kwant.Builder(sym_left_lead) for j in xrange(W): left_lead[lat(0, j)] = 4 * t+pot if j > 0: left_lead[lat(0, j), lat(0, j - 1)] = -t left_lead[lat(1, j), lat(0, j)] = -t sys.attach_lead(left_lead) sym_right_lead = kwant.TranslationalSymmetry((a, 0)) right_lead = kwant.Builder(sym_right_lead) for j in xrange(W): right_lead[lat(0, j)] = 4 * t if j > 0: right_lead[lat(0, j), lat(0, j - 1)] = -t right_lead[lat(1, j), lat(0, j)] = -t sys.attach_lead(right_lead) # Modify the scattering region del sys[lat(85, 2)] del sys[lat(87, 2)] del sys[lat(86, 3)] del sys[lat(25, 37)] del sys[lat(27, 37)] del sys[lat(26, 38)] sys = sys.finalized() return sys

2 3