Mailman 3 March 2020 - 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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To to separate spins, electron and hole
by Khani Hosein 22 Jun '21

22 Jun '21

Dear all, I am using Kwant to study a system with both spins and electron and hole, so the hopping matrix is 4X4. Now I want to know how to separate the spins, electron and hole. I study this through the "2.6. Superconductors: orbital degrees of freedom, conservation laws and symmetries" smatrix = kwant.smatrix(syst, energy) data.append(smatrix.submatrix((0, 0), (0, 0)).shape[0] -smatrix.transmission((0, 0), (0, 0)) +smatrix.transmission((0, 1), (0, 0))) I do not find more information about this for kwant. For my system, if I have 3 leads, and the hopping and onsite energy is set in this order: (e↑,0,0,0)- spin up electron,first row of the 4X4matrix (0,e↓,0,0)- spin down electron,second row of the 4X4matrix (0,0, h↑ ,0)- spin up hole,third row of the 4X4matrix (0,0, 0, h ↓ )- spin down hole,fourth row of the 4X4matrix I want to obtain the transmissions from 0 →2. smatrix = kwant.smatrix(syst, energy) The transmission from h↑ to e↑ is: smatrix.transmission((2, 1), (0, 2)) The transmission from h ↓ to e↑ is: smatrix.transmission((2, 1), (0, 3)) Is my understanding correct? Thanks very much in advance! Hosein Khani

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Stability of the retarded Green's function calculation
by Joseph Weston (Aquent LLC - Canada) 26 Mar '20

26 Mar '20

Hello Kwantoptians, I have a question regarding the speed/stability of computing the retarded Green's function of a transport setup using Kwant. In the Kwant source-code [1] it is noted that using `kwant.greens_function` is "often slower and less stable than the scattering matrix calculation". I was wondering if you could provide me with some references for this assertion. I have perused the Kwant paper [2] and (part I of) the thesis of Michael Wimmer [3] and cannot find any mention of a speed/stability comparison of the algorithm implemented by `kwant.greens_function` vs `kwant.smatrix`. My understanding is that in both cases a linear system (LHS) is constructed and solved for different right hand-sides (RHS) using out of the box linear solvers. The solution for each RHS corresponds to one column of the retarded Green's function / extended scattering matrix respectively. The difference between `kwant.greens_function` and `kwant.smatrix` is then the following. In the former case the leads are taken into account by added the retarded self-energy to the LHS and the RHSs are indicator vectors for the sites of the lead/scattering region interface, which corresponds to a "unit impulse" boundary condition. In the latter case the linear system is "extended" so as to include extra unknowns that correspond to the scattering amplitudes, and the RHSs are indicator vectors with the 1's in this "extended" part, which corresponds to a "single incoming mode" boundary condition. It seems to me that the salient difference is in the boundary conditions, and I do not have a good intuition as to why one set of boundary conditions would make the linear system easier/harder to solve. Happy Kwanting! Joe [1]: https://gitlab.kwant-project.org/kwant/kwant/-/blob/master/kwant/solvers/co… [2]: https://iopscience.iop.org/article/10.1088/1367-2630/16/6/063065/pdf [3]: https://epub.uni-regensburg.de/12142/

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3D Current Density
by Ousmane LY 25 Mar '20

25 Mar '20

Dear Naveen, if you want to use a 2d system to represent local properties of a 3d system, the former shouldn’t be considered as a physical system. Therefore, its onsite and hopping values are not relevant and shouldn’t alter your calculations in any sens. Even attaching leads for the 2d system may not be necessary. However, you have to be sure that the 2d system represents well the 2d plane of the underlying 3d geometry. This would concern mainly the number of sites and links. You may have a look to the illustration below. Happy kwanting, Ousmane, ############################################################ import kwant def make_3d(L=5,W=5,H=5): # make a 3d system sys=kwant.Builder() lat=kwant.lattice.cubic(norbs=1) sys[(lat(x,y,z) for x in range(L) for y in range(W) for z in range(H))]=0 sys[lat.neighbors()]=-1 lead=kwant.Builder(kwant.TranslationalSymmetry((-1,0,0))) lead[(lat(0,y,z) for y in range(W) for z in range(H))] = -1 lead[lat.neighbors()]=-1 sys.attach_lead(lead) return sys.finalized() def make_2d(W=5,H=5): # make an auxiliary 2d system sys=kwant.Builder() lat=kwant.lattice.square() sys[(lat(y,z) for y in range(W) for z in range(H) )]= " " sys[lat.neighbors()]= " " return sys.finalized() def main(): # calculate the current at a given (y,z) plane given_x=0 # e.g def where(site1,site2): return site1.pos[0]==given_x and site2.pos[0]==given_x psi=kwant.wave_function(make_3d(),energy=0)(0)[0] # lowest mode of lead 0 current=kwant.operator.Current(make_3d(), where=where)(psi) # plot the current using the auxiliary 2d system kwant.plotter.current(make_2d(), current) if __name__=='__main__': main() ##################################################################

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adding spin degrees of freedom to BdG form in the superconductor.py in Kwant
by Sayandip Dhara 24 Mar '20

24 Mar '20

Hello, I want to do a transport calculation for a 1D/2D nanowire supporting Majorana fermions. So for that, I need to add spin and changing magnetic field and spin-orbit coupling. In the section for superconductors (example 2.6 )the BdG hamiltonian is expressed as a 2x2 system with tau matrices. Now if I want to add spin and spin-orbit coupling H_0 I have to write the onsite terms as 2x2 matrices (pauli matrices). So my question is how can I do that in this BdG formalism in Kwant. Also, I want to know a way to plot the differential conductance with the voltage of the leads. Thanks, Sayandip Dhara University of Central Florida

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3D Current Density
by Naveen Yadav 23 Mar '20

23 Mar '20

Dear KWANT developers, I want to plot the current density for a 3D model with finite width in x and y direction. Below is the energy dispersion of the model Hamiltonian. To calculate the current density profile, I have assumed scattering region with a width of 60 unit cells in z-direction and leads with different onsite and same hoppings (as that of scattering region) are attached. For better understanding the code is attached below. For plotting current density, I have used 2D slices of y-z plane. Now my question is that, while building the second system, what onsite and Hopping terms should I have to use? Because I have already build a 3D system with particular onsite and hopping terms. Can I use onsite =0 for the second system? and what should be the hoppings for the second system? can I use simply lat2.neighbors() for the second system? I tried it in different ways, it gives me different results. I don't know which one correct. please suggest me the correct way. [image: image.png] *import kwantfrom numpy import exp,sin,linspace,sqrt, cos, piimport matplotlib as mplfrom matplotlib import pyplot as pltimport tinyarrayimport pandas as pd* *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]])lat=kwant.lattice.cubic(norbs=2)L=10W=80#H=60def onsite(site): return (1.0 * cos(1.05) + 2 * 1.0) * sigma_z* *def onsite_s(site): return (1.0 * cos(1.05) + 2 * 1.0) * sigma_z - 0.75 * sigma_0def hoppingx(site0, site1): return (-0.5 * 1.0 * sigma_z - 0.5 * 1j * 1.0 * sigma_x)def hoppingy(site0, site1): return -0.5 * 1.0 * sigma_z - 0.5 * 1j * 1.0 * sigma_ydef hoppingz(site0, site1): y = site1.pos[1] return (-0.5 * 1.0 * sigma_z - 0.5 * 1j * 0.25 * sigma_0) * exp(-2 * pi * 1j * 0.001 * 1.0 * (y-40))sys=kwant.Builder()sys[(lat(z,y,x) for z in range(0,10) for y in range(W) for x in range(L))]=onsite_ssys[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingxsys[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingysys[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingzsys[(lat(z,y,x) for z in range(10,70) for y in range(W) for x in range(L))]=onsitesys[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingxsys[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingysys[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingzsys[(lat(z,y,x) for z in range(70,80) for y in range(W) for x in range(L))]=onsite_ssys[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingxsys[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingysys[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingz* *# The scattering region 0 < W_z < 10 and 60 < W_z > 70 have onsite energy of leads.* *lead=kwant.Builder(kwant.TranslationalSymmetry((-1,0,0)))lead[(lat(0,y,x) for y in range(W) for x in range(L))] = onsite_slead[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hoppingzlead[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hoppingylead[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hoppingxsys.attach_lead(lead)sys.attach_lead(lead.reversed())kwant.plot(sys)sysf=sys.finalized()Sites=list(sysf.sites)#print(Sites)wave=kwant.wave_function(sysf, energy=0.1)psi=wave(0)[0]#df = pd.DataFrame(Sites)#df.to_csv('data2.csv',index=False, header=False)def Plot(x,psi): def cut(site0,site1): return site0.pos[2]==x and site1.pos[2]==x J_0 = kwant.operator.Current(sysf, where=cut) current = J_0(psi) lat2=kwant.lattice.square(norbs=2) sys2=kwant.Builder() sys2[(lat2(site.pos[0],site.pos[1]) for site in Sites if site.pos[2]==x)]=onsite* * #sys2[(lat2(site.pos[0],site.pos[1]) for site in Sites if site.pos[2]==x)]=0* * sys2[kwant.builder.HoppingKind((1, 0), lat2, lat2)] = hoppingz sys2[kwant.builder.HoppingKind((0, 1), lat2, lat2)] = hoppingy* * #sys2[lat2.neighbors()]=-sigma_0* * #lead2=kwant.Builder(kwant.TranslationalSymmetry((-1, 0))) #lead2[(lat2(site.pos[0], site.pos[1]) for site in Sites if site.pos[2]==x)]=onsite_s #lead2[kwant.builder.HoppingKind((1, 0), lat2, lat2)] = hoppingz #lead2[kwant.builder.HoppingKind((0, 1), lat2, lat2)] = hoppingy #sys2.attach_lead(lead2) #sys2.attach_lead(lead2.reversed()) sys2f=sys2.finalized() #kwant.plot(sys2f) kwant.plotter.current(sys2f, current)for i in range(10): Plot(i,psi)* -- With Best Regards Naveen Yadav Ph.D Research Scholar Deptt. Of Physics & Astrophysics University Of Delhi.

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How to obtain the retarded Green's function for a lead
by Joseph Weston (Aquent LLC - Canada) 16 Mar '20

16 Mar '20

Good day, I have a question about how best to obtain the retarded Green's function for a lead in Kwant. Kwant already has a way of calculating the retarded self-energy of a lead (the 'selfenergy' method of 'StabilizedModes' [1]), however there is no equivalent routine for the retarded Green's function. As far as I am aware one may obtain the self-energy from the Green's function by sandwiching it between a V (inter-cell hopping) and a V^\dagger, and I was wondering if there is a reason why there is not a routine that returns the Green's function directly. One possibility is because the modes, as calculated by Kwant, are (in general) in a weird basis related to the singular value decomposition of V, and only a part of the SVD is stored with the stabilized modes, which makes it impossible to "get out" the modes in the tight-binding basis as required for the Green's function. Is my understanding correct, or am I missing something? Thanks, Joe [1]: https://gitlab.kwant-project.org/kwant/kwant/-/blob/master/kwant/physics/le…

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Kwant-discuss March 2020