Mailman 3 July 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>

4 4

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

4 4

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))

3 3

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

3 3

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

4 3

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

3 4

Migrating spin_orbit.py to real unit system
by Nathaniel Gregory Seil 06 Aug '20

06 Aug '20

Hello, I would like to model a 2DEG under a magnetic field and have been investigating the spin_orbit.py code. I would like to modify this code so that it utilises real units and models upon a to-scale lattice. However, I am a little bit unclear as to how to progress. I noticed in the base code we have a = 1, t = 1 and hbar = 2 (seen from the Pauli spin matrices which normally have hbar/2 as a coefficient in front). To represent the real life scenario of a lattice with constant a = 5E-10 m, would we do the following: * a = 5E-10 m * hbar = 1.05457E-34 J s (standard SI units) * m_electron = 9.109E-31 kg (standard SI units) * t = hbar ^ 2 / (2 * m_electron * a^2) and then just substitute these values into the rest of the code? I get the result of 0 conductivity for the lead voltages -0.046 to 0.105V (code shows "energies=[0.01 * i*t - 0.3*t for i in range(100)]"). I also get this for -0.0046 to 0.0105V and for -0.46 - 1.05V. Do these results seem right? I have attached my code with these substitutions and the results. Many thanks, Nathaniel Seil

2 5

Peierls phase from kwant.physics.magnetic_gauge
by Paweł Wójcik 31 Jul '20

31 Jul '20

Dear KWANT community, Recently I have used KWANT to simulate electronic transport in T-junction with the perpendicular magnetic field applied to the nanostructure. As well known, the inclusion of the orbital effects in this case (by the peierls substitution) is not easy due to appropriate gauge which should be applied for the vertical leads. So I decided to use the newest functionality of KWANT included in kwant.physics.magnetic_gauge. I started testing this function from a simple nanowire (NW) with two leads applied on both sides of NW and the magnetic field perpendicular to NW. When calling ham=sys.hamiltonian_submatrix(params=params), similarly as in documentation, everything works OK. The problem appears when I am trying to calculate transport and call the function smatrix = kwant.smatrix(sys, energy=energy, params=params). Then, an following error occurs i = syst.id_by_site[a] j = syst.id_by_site[b]# We only store *either* (i, j) *or* (j, i). If not presentKeyError: Site(kwant.lattice.Monatomic([[75.58904535685008, 0.0], [0.0, 75.58904535685008]], [0.0, 0.0], '', None), array([-1, -9])) The above exception was the direct cause of the following exception: And I don't know how is the reason of this error. Below I paste my code nw = SimpleNamespace(\ W = 10, L = 100, dx = nm2au(4), m = 0.015, B = T2au(.5), ) def make_system_transport_gauge(nw): W=nw.W L=nw.L dx=nw.dx m=nw.m B=nw.B def onsite(sitei,peierls): x,y=sitei.pos t=1.0/(2.0*nw.m*nw.dx*nw.dx) return 4*t def hopping(sitei, sitej, peierls): xi,yi=sitei.pos xj,yj=sitej.pos t=1.0/(2.0*nw.m*nw.dx*nw.dx) return -t*peierls(sitei,sitej) lat=kwant.lattice.square(dx) sys = kwant.Builder() sys[(lat(i,j) for i in range(L) for j in range(-W+1,W))]=onsite sys[lat.neighbors()] = hopping syml = kwant.TranslationalSymmetry((-dx, 0)) l_lead=kwant.Builder(syml) l_lead[(lat(0,y) for y in range(-W+1,W))]=onsite l_lead[lat.neighbors()]=hopping l_lead.substituted(peierls='peierls_llead') sys.attach_lead(l_lead) symr = kwant.TranslationalSymmetry((dx, 0)) r_lead=kwant.Builder(symr) r_lead[(lat(0,y) for y in range(-W+1,W))]=onsite r_lead[lat.neighbors()]=hopping r_lead.substituted(peierls='peierls_rlead') sys.attach_lead(r_lead) sys = sys.finalized() return sys def B_syst(pos): return nw.B sys=make_system_transport_gauge(nw) gauge = kwant.physics.magnetic_gauge(sys) peierls_sys, peierls_llead, peierls_rlead = gauge(B_syst, B_syst, B_syst) params = dict(t=1, peierls=peierls_sys, peierls_llead=peierls_llead, peierls_rlead=peierls_rlead) ham=sys.hamiltonian_submatrix(params=params) #this works #I tried to calculate transport energy=eV2au(3e-3) smatrix = kwant.smatrix(sys, energy=energy, params=params) #this generates an error I would be grateful for any help or hint what is wrong. Best regards, Pawel Wojcik

2 3

Conductivity in 2D system with Zeeman and Rashba spin-orbit interaction
by anna.krzyzewska＠amu.edu.pl 29 Jul '20

29 Jul '20

Dear Kwant community, I would like to calculate the DOS and conductivity in 2D system with Zeeman and Rashba spin-orbit interaction in a square lattice. Regarding the chapter 2.3.1 in KWANT tutorial I defined the function make_system(). Then, according to the chapter 2.9 in the tutorial, I used the kwant.kpm.conductivity() method to calculate the longitudinal and transverse conductivity in the system. But I see that my results are wrong and I do not understand some things: 1. question: in chapter 2.3.1 we did not need to define the norbs parameter but I suppose that KWANT "knows" that we have here two orbitals per site (considering the spin) because we use the Pauli matrices. But, to obtain some results using function kwant.kpm.conductivity(), I see that the argument norbs=2 in the system deifinition is required (without it I get the error: 'NoneType' object cannot be interpreted as an integer). Thus, my quesion is: how does KWANT interpret my system if I add or not this argument (nobrs=2)? I think this could be the problem causing my wrong results. 2. question: To use the method kwant.kpm.conductivity() we don't need to attach the leads. Maybe it is obvious and trivial quesion, but I do not understand how we can calculate the conductivity (eg. induced by the external electric field) in a closed system? And what about the boundary conditions? Below I present the whole code. #-------------------------------------------------------------------------------------- #-------------------------------------------------------------------------------------- import kwant from matplotlib import pyplot import tinyarray # we define the identity matrix and Pauli matrices 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]]) def make_system(t=1.8, alpha=0.1, e_z=0.0018, W=30, L=30, a=1): lat = kwant.lattice.square(norbs=2) # norbs - the number of orbitals per site syst = kwant.Builder() syst[(lat(x,y) for x in range(L) for y in range(W))] = 4*t*sigma_0 + e_z*sigma_z # Zeeman energy adds to the on-site term syst[kwant.builder.HoppingKind((a,0), lat, lat)] = -t*sigma_0 - 1j*alpha*sigma_y/(2*a) # hoppings in x-direction; (a,0) means hopping form (i,j) to (i+1,j) syst[kwant.builder.HoppingKind((0,a), lat, lat)] = -t*sigma_0 + 1j*alpha*sigma_x/(2*a) # hoppings in y-direction; (0,a) means hopping form (i,j) to (i,j+1) return syst def plot_dos_and_curves(dos, labels_to_data): pyplot.figure() pyplot.fill_between(dos[0], dos[1], label="DoS [a.u.]", alpha=0.5, color='gray') for label, (x, y) in labels_to_data: pyplot.plot(x, y, label=label, linewidth=2) pyplot.legend(loc=2, framealpha=0.5) pyplot.xlabel("energy [t]") pyplot.ylabel("$\sigma [e^2/h]$") pyplot.show() def main(): syst = make_system() kwant.plot(syst); # check if the system looks as intended syst = syst.finalized() spectrum = kwant.kpm.SpectralDensity(syst, rng=0, num_vectors=30) # object that represents the density of states for the system energies, densities = spectrum() # component 'xx' cond_xx = kwant.kpm.conductivity(syst, alpha='x', beta='x', rng=0) # component 'xy' cond_xy = kwant.kpm.conductivity(syst, alpha='x', beta='y', rng=0) energies = cond_xx.energies cond_array_xx = np.array([cond_xx(e, temperature=0.01) for e in energies]) cond_array_xy = np.array([cond_xy(e, temperature=0.01) for e in energies]) plot_dos_and_curves( (spectrum.energies, spectrum.densities * 8), # why multiplied by 8? [ (r'Longitudinal conductivity $\sigma_{xx} / 4$', (energies, cond_array_xx.real / 4)), (r'Hall conductivity $\sigma_{xy}$', (energies, cond_array_xy.real))], ) # standard Python construct to execute 'main' if the program is used as a script if __name__ == '__main__': main() #-------------------------------------------------------------------------------------- #-------------------------------------------------------------------------------------- Regards, Anna Krzyzewska

2 7

(no subject)
by Prof. CHAN Kwok Sum 29 Jul '20

29 Jul '20

Dear Kwant developer, I want to define a spatially varying current operator which depends on the two sites of a bond. Can this be done giving the “onsite” a function in the current operator definition. According to the manual, the function for onsite has the signature of a Hamiltonian on-site function, does it mean the function can only have one site as its input and cannot have two sites of a bond as input? If “onsite” cannot do the job, any suggestion of how to solve the problem? Regards, 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.

3 4