Mailman 3 March 2018 - Kwant-discuss

Gate-defined Nanowire Quantum Dot
by Jonathan Fields Sept. 15, 2022

Sept. 15, 2022

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>

4 4

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

Spin polarized transport calculation
by L.M J March 6, 2022

March 6, 2022

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

3 3

How to caculate hall resistance
by ZHANG Bing Sept. 12, 2021

Sept. 12, 2021

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

4 3

Visualize current through a 2D cut for a 3D system
by elchatz＠auth.gr May 4, 2018

May 4, 2018

Hello everyone, Is there currently way to visualize the current through a 2D cut for a 3D system that does not involve messing with the kwant code? I have a 3D system and would like to see the differences in the current when I switch some specific hoppings on and off. Regards, -- Dr. Eleni Chatzikyriakou Computational Physics lab Aristotle University of Thessaloniki elchatz(a)auth.gr - tel:+30 2310 998109

4 12

Fwd: Visualize current through a 2D cut for a 3D system
by elchatz＠auth.gr March 28, 2018

March 28, 2018

So, for example, I can print the on-site energies or the hopping integrals of the sites I want using: ----- print(kwant_sys[lattice.sublattices[4](0,0,-1)) and print(kwant_sys[lattice.sublattices[4](0,0,-1), lattice.sublattices[6](0,0,-1)]) ----- respectively. I can get the tags and families of the sites as such: ----- >>> for x in kwant_sys.sites(): ... print(x.tag, ',', x.family) ... [0 0 -1] , <Monatomic lattice 0> [0 0 -1] , <Monatomic lattice 1> .... ------ However, I am still unsure about the 'where' argument ------------------------------------------------------------------------------------------------- >>> J = kwant.operator.Current(kwant_sys, >>> where=kwant_sys[lattice.sublattices[4](0,0,-1), >>> lattice.sublattices[6](0,0,-1)]) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "kwant/operator.pyx", line 883, in kwant.operator.Current.__init__ File "kwant/operator.pyx", line 217, in kwant.operator._normalize_hopping_where File "kwant/operator.pyx", line 218, in genexpr ValueError: need more than 1 value to unpack >>> J = kwant.operator.Current(kwant_sys, >>> where=(lattice.sublattices[4](0,0,-1), >>> lattice.sublattices[6](0,0,-1))) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "kwant/operator.pyx", line 883, in kwant.operator.Current.__init__ File "kwant/operator.pyx", line 223, in genexpr File "kwant/operator.pyx", line 223, in genexpr AttributeError: 'Builder' object has no attribute 'graph' -------------------------------------------------------------------------------------------------- Eleni ----- Forwarded message from elchatz(a)auth.gr ----- Date: Mon, 26 Mar 2018 17:19:18 +0300 From: elchatz(a)auth.gr Subject: Re: [Kwant] Visualize current through a 2D cut for a 3D system To: kwant-discuss(a)kwant-project.org Hello Anton, Thank you very much for your reply. I am having a hard time with the "where" argument in kwant.operator.Current. My system is not finalized. How can I find the sites associated with the hopping? By their tag? Their position? Do you have a piece of code for that? Note that I am using TBModels and Wannier90 and the lattice has one sublattice for each WF (or site, or orbital). norb is not defined, but I see that for x in lattice.sublattices: x.norbs=1 works and looks correct. Regards, Eleni Quoting Anton Akhmerov <anton.akhmerov+kd(a)gmail.com>: > Dear Eleni, > > I think this should be doable with not too much of low level work. > * Firstly you probably will want to select only the hoppings that are > close to the 2D cut. You can do this by using "where" parameter to > Current operator (see > https://kwant-project.org/doc/1/reference/generated/kwant.operator.Current#…) > * After computing the current across those hoppings, you would need to > call kwant.plotter.interpolate_current. This function only does the > interpolation, but no plotting. Having that interpolation in 3D you > need to slice it and probably take a projection onto 2D plane. > * Finally, the resulting 2D array you can feed to kwant.plotter.streamplot > > kwant.plotter.current is essentially a straightforward combination of > kwant.plotter.interpolate_current and kwant.plotter.streamplot > > Also: your problem sounds like a useful thing to do. Please let me > know if you succeed, and what you ended up doing (or if you face any > further problems as well). > > Best, > Anton > > On Fri, Mar 23, 2018 at 5:22 PM, <elchatz(a)auth.gr> wrote: >> Hello everyone, >> >> Is there currently way to visualize the current through a 2D cut for a 3D >> system that does not involve messing with the kwant code? >> >> I have a 3D system and would like to see the differences in the current when >> I switch some specific hoppings on and off. >> >> >> Regards, >> >> >> >> -- >> Dr. Eleni Chatzikyriakou >> Computational Physics lab >> Aristotle University of Thessaloniki >> elchatz(a)auth.gr - tel:+30 2310 998109 >> -- Dr. Eleni Chatzikyriakou Computational Physics lab Aristotle University of Thessaloniki elchatz(a)auth.gr - tel:+30 2310 998109 ----- End forwarded message ----- -- Dr. Eleni Chatzikyriakou Computational Physics lab Aristotle University of Thessaloniki elchatz(a)auth.gr - tel:+30 2310 998109

2 2

modes and current density
by Nicolas Moreau March 27, 2018

March 27, 2018

Hello everyone, I want to compute and plot the current density in a graphene sample. I logically use kwant.operator.Current(syst) and then kwant.plotter.current(syst, current) to plot the current density in the sample. However, the main example in the kwant documentation only uses the wavefunction associated to the first mode to compute J: ________________________________ wf = kwant.wave_function(syst, energy=-0.3) J0 = kwant.operator.Current(syst) wf_left = wf(0)[0] current = J0(wf_left) ________________________________ In my case, I have a system with more than one mode. I have found some lines that consider all the modes (at the end of this page : https://kwant-project.org/doc/dev/tutorial/operators ) 2.7. Computing local quantities: densities and currents ...<https://kwant-project.org/doc/dev/tutorial/operators> kwant-project.org 2.7. Computing local quantities: densities and currents¶ In the previous tutorials we have mainly concentrated on calculating global properties such as conductance ... ________________________________ wf_left = wf(0) J_m_left = sum(J_m_bound(p) for p in wf_left) J_z_left = sum(J_z_bound(p) for p in wf_left) ________________________________ But this is equivalent to calculate the expression [cid:96ff5a32-4b82-4677-ab3d-04d9e9e1233e] which is simply the sum over the current coming from each mode. However, I would have rather use the following expression to find a more general expression of the current density [cid:a1236ff5-dbcc-445a-98b4-712fc485334f] What is therefore the good way to find J? I have not found more informations in the Kwant documentation. Best regards, Nicolas

2 1

Coordinates of spatial greens function
by yuhao kang March 21, 2018

March 21, 2018

Dear Kwant developer, I'm trying to calculate the greens function as follows: gf = kwant.greens_function(sys,0).submatrix(1,0) It returns a 18*18 matrix. Does it mean the green function between 18 points in lead0 to 18 points in lead1? However, when I plot this system, there is only 9 points at each lead. If the base used here are not the lattice point? Can I get the coordinates of the 18 points in both leads? So I can know the green function between two designated positions. Thank you. Best, Yuhao Here is my code. ## import numpy as np import kwant m, t=0.4, -0.133 dis=0 a=1 lat = kwant.lattice.general([[a, 0], [a/2, a*np.sqrt(3)/2]], [[0, 0], [0, a/np.sqrt(3)]]) A,B=lat.sublattices sys=kwant.Builder() s0=np.identity(2) sx=np.array([[0,1],[1,0]]) sz=np.array([[1,0],[0,-1]]) def disk(pos): x,y=pos return abs(y)<2 and np.sqrt(3)*x-1<y<np.sqrt(3)*x+5 def edge(pos): x,y=pos return abs(y)<2 and np.sqrt(3)*x-0.1<y<np.sqrt(3)*x+0.58 def edge1(pos): x,y=pos return abs(y)<2 and np.sqrt(3)*x-0.1<y<np.sqrt(3)*x+0.58 def hopp(site1,site2): x1,y1=site1.pos x2,y2=site2.pos if (y1+0.1*a)*(y2+0.1*a)>0 and y1+0.1*a<0: return -t*s0 else: return np.zeros([2,2]) def hopp2(site1,site2): x1,y1=site1.pos x2,y2=site2.pos if (y1+0.1*a)*(y2+0.1*a)>0 and y1+0.1*a<0: return t*s0 else: return np.zeros([2,2]) def onsite(site): x,y=site.pos if (x-25*a)**2+(y-9*a)**2<8: return -m*sz+0*sx elif y>-0.1*a: return m*sz+0*sx else: return np.zeros([2,2]) sys[lat.shape(disk,(0,0))]=onsite sys[lat.neighbors()]=-s0*2/3 sys[A.neighbors()]=hopp sys[B.neighbors()]=hopp2 lead0=kwant.Builder(kwant.TranslationalSymmetry((-a,0))) lead0[lat.shape(edge,(0,0))]=onsite lead0[lat.neighbors()]=-s0*2/3 lead0[A.neighbors()]=hopp lead0[B.neighbors()]=hopp2 sys.attach_lead(lead0) sys.attach_lead(lead0.reversed()) kwant.plot(sys, fig_size=(30, 12)) sys=sys.finalized() gf=kwant.greens_function(sys,0).submatrix(1,0) ##

2 1

Profiles of incoming transverse waveguide modes
by amrita chapagain March 14, 2018

March 14, 2018

Thank You Adel, Thank you for helping on previous issue. I have another curiosity here. Right now all incoming flux on lead 0 is all modes from 1 to N. The incoming flux corresponds to summation from 1 to N (i.e ), I just want first 3 modes only. I have attached a picture here what I want to do it. Can I achieve such condition in kwant from finding wavefunction from kwant solver? Here I have attached a code to find wavefunction. import kwant import numpy as np from matplotlib import pyplot from numpy import sqrt from math import * #====================================================================== # Define the shape ------------------- #====================================================================== def shape(pos): x, y = pos return (np.abs(2.0*x)<=L)&(np.abs(y)<W1/2.0) #---------------------------------------------------------------------- #====================================================================== def wshape(pos): x, y = pos return (np.abs(2.0*x)<=L)&(np.abs(y)<W1/2.0) #---------------------------------------------------------------------- a = 1; t = 1; E0L = 0*t L = 10; # Length of the W1 = 30; # Width on the left # Define geometry #-------------------------------------------------------------------------- sys0 = kwant.Builder() lat = kwant.lattice.square(a) #-------------------------------------------------------------------------------------------- # Define onsite energies and couplings #---------------------------------------------------- sys0[lat.shape(wshape,(0,0))] = E0 # To make all sites the same sys0[lat.neighbors()] = t #-------------------------------------------------------------------------------------------- # Left lead #---------------------------------------------------------------------------------- left_lead = kwant.Builder(kwant.TranslationalSymmetry([-1,0])) left_lead[(lat(0,y) for y in range(int(-W1/2+1),int(W1/2)))] = E0L left_lead[lat.neighbors()] = t sys0.attach_lead(left_lead); #-------------------------------------------------------------------------------------------- # Right lead #--------------------------------------------------------------------------------- right_lead = kwant.Builder(kwant.TranslationalSymmetry([1,0])) right_lead[(lat(0,y) for y in range(int(-W1/2+1),int(W1/2)))] = E0L right_lead[lat.neighbors()] = t sys0.attach_lead(right_lead); #--------------------------------------------------------------------------------------------- sys = sys0.finalized() kwant.plot(sys); wf = kwant.solvers.default.wave_function(sys,E,check_hermiticity=False); wf0 = wf(0); wf1 = wf(1); def plot_conductance(sys, energies): data = [] for energy in energies: smatrix = kwant.smatrix(sys, energy) data.append(smatrix.transmission(1, 0)) #transmission from left 2 leads to right lead pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() energies=[-3+i*0.02 for i in range(100)] plot_conductance(sys,energies)

2 1

Profiles of incoming transverse waveguide modes
by amrita chapagain March 14, 2018

March 14, 2018

Hi all, Thank you for addressing my previous query. I am trying to plot first few incoming transverse waveguide modes on lead 0. I am wondering whether it is possible to plot all incoming modes on Lead 0. kwant.physics.modes gives me PropagatingModes and Stabilized modes but I want to select just first few modes of incoming flux. Here is the sample code of my system import kwant import numpy as np from matplotlib import pyplot from numpy import sqrt from math import * #====================================================================== # Define the shape ------------------- #====================================================================== def shape(pos): x, y = pos return (np.abs(2.0*x)<=L)&(np.abs(y)<W1/2.0) #---------------------------------------------------------------------- a = 1; t = 1; E0L = 0*t L = 10; # Length of the W1 = 30; # Width on the left # Define geometry -------------------------------------------------------------------------- sys0 = kwant.Builder() lat = kwant.lattice.square(a) #-------------------------------------------------------------------------------------------- # Define onsite energies and couplings ---------------------------------------------------- sys0[lat.shape(wv_shape,(0,0))] = E0 # To make all sites the same sys0[lat.neighbors()] = t #-------------------------------------------------------------------------------------------- # Left lead ---------------------------------------------------------------------------------- left_lead = kwant.Builder(kwant.TranslationalSymmetry([-1,0])) left_lead[(lat(0,y) for y in range(int(-W1/2+1),int(W1/2)))] = E0L left_lead[lat.neighbors()] = t sys0.attach_lead(left_lead); #-------------------------------------------------------------------------------------------- # Right lead --------------------------------------------------------------------------------- right_lead = kwant.Builder(kwant.TranslationalSymmetry([1,0])) right_lead[(lat(0,y) for y in range(int(-W1/2+1),int(W1/2)))] = E0L right_lead[lat.neighbors()] = t sys0.attach_lead(right_lead); #--------------------------------------------------------------------------------------------- sys = sys0.finalized() kwant.plot(sys); def plot_conductance(sys, energies): data = [] for energy in energies: smatrix = kwant.smatrix(sys, energy) data.append(smatrix.transmission(2, 0)+smatrix.transmission(1,2) ) #transmission from left 2 leads to right lead pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() energies=[-3+i*0.02 for i in range(100)] plot_conductance(sys,energies)

2 1