Mailman 3 September 2022 - Kwant-discuss

Realistic values of hopping parameters
by Yu Li 19 Oct '23

19 Oct '23

Dear all, Now I’m studying some transport property based on multilayered system, with nonmagnetic layer/ferromagnetic layer (NM/FM) stack. May I ask when creating a tight binding system, and setting hoppings by the following function: syst[(lat(1, 0), lat(1, 1))] = t, how to associate the hopping with a realistic system? Such as hoppings between FM and FM atoms, or between FM and NM atoms? I found in many literatures people just simply put t=1, but I assume the value should depend on the type of atoms. My another question is, is it possible to define more than one type of hopping between two sites? For the tight-binding approximation of p electrons, there are two types of hopping integrals: Vppσ, and Vppπ, so I’m wondering if it’s possible to simulate the effect of both hoppings? Thanks a lot for reading my questions! Cheers, Yu Li

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Defining uniaxial strain in specified region of graphene
by shrushti.tapar07＠gmail.com 20 Jan '23

20 Jan '23

I want to create the graphene strained superlattice-like structure, having uniaxial strain defined at the specified region. Please, someone can suggest to me how to define the strained region and interface between unstrained and strained graphene regions. Thanks in Advance Shrushti

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Does Kwant use Transfer Matrix like Methods or ED?
by Kartikeya Arora 27 Sep '22

27 Sep '22

I tried to read the documentations and everything but got a bit confused. Here's my question and the task I want to complete. I want to simulate a 1d Anderson Model and calculate either Localization Length or Conductance for different system sizes. Now, using normal ED, I can only go up to 5000 system size in 24 hours (100 disorder realizations). Now, I want to calculate quantities like Localization Length/Conductance as a function of system size. At your best estimate, How much system size I'll be able to reach and in what time? And how should I go about coding it? Thanks, Kartikeya

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Re: How the spin is positioned in the Smatrix?
by Diego Bruno 27 Sep '22

27 Sep '22

Can you disponibilize the code?

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Calculation of Andreev Spectrum for NS junction in KWANT
by Subhadeep Chakraborty 26 Sep '22

26 Sep '22

Hi Kwant Users ! Can anyone tell me if one could obtain the andreev spectrum for an NS junction using KWANT. ___________________________________ It would be a great a help if someone shares a basic idea regarding how to do it ? Thanks in advance,

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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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Issue in understanding how conservation law works for separating electron and hole blocks
by Binayyak Roy 15 Sep '22

15 Sep '22

Hi, *I am trying to build a 2D quantum wire with both Rashba spin coupling and superconductivity*. I have been building on the code of superconductivity already mentioned in the kwant documentation pdf and added the rashba coupling terms to the code. However when I add the asymmetric terms of the rashba spin coupling it creates some sort of an error giving me an empty scattering matrix. What could be causing this? After going through the documentation I realize that it's probably got something to do with the fact that I have the wrong conservation law. I am not sure how to set that up? Sincerely, Binayyak I have attached my code below. import kwant import tinyarray import numpy as np # For plotting from matplotlib import pyplot tau_0 = tinyarray.array([[1, 0], [0, 1]]) tau_x = tinyarray.array([[0, 1], [1, 0]]) tau_y = tinyarray.array([[0, -1j], [1j, 0]]) tau_z = tinyarray.array([[1, 0], [0, -1]]) def make_system( a=1, W=10, L=10, barrier=1.5, barrierpos=(3, 4), mu=0.4, Delta=0.1, Deltapos=4, t=1.0, E_z=1e-1, # zeeman splitting field alpha=1e-3, # rashba coupling coefficient phs=True, ): lat = kwant.lattice.square(norbs=2) syst = kwant.Builder() #### Define the scattering region. #### # The superconducting order parameter couples electron and hole orbitals # on each site, and hence enters as an onsite potential. # The pairing is only included beyond the point 'Deltapos' in the scattering region. syst[(lat(x, y) for x in range(Deltapos) for y in range(W))] = ( 4 * t - mu + E_z ) * tau_z syst[(lat(x, y) for x in range(Deltapos, L) for y in range(W))] = ( 4 * t - mu + E_z ) * tau_z + Delta * tau_x # The tunnel barrier syst[(lat(x, y) for x in range(barrierpos[0], barrierpos[1]) for y in range(W))] = ( 4 * t + barrier - mu + E_z ) * tau_z # Hoppings syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = -t * tau_z + 1j * alpha * tau_y syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t * tau_z - 1j * alpha * tau_x #### Define the leads. #### # Left lead - normal, so the order parameter is zero. sym_left = kwant.TranslationalSymmetry((-a, 0)) # Specify the conservation law used to treat electrons and holes separately. # We only do this in the left lead, where the pairing is zero. lead0 = kwant.Builder(sym_left, conservation_law=-tau_z, particle_hole=tau_y) lead0[(lat(0, j) for j in range(W))] = (4 * t - mu + E_z) * tau_z lead0[kwant.builder.HoppingKind((1, 0), lat, lat)] = -t * tau_z + 1j * alpha * tau_y lead0[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t * tau_z - 1j * alpha * tau_x # Right lead - superconducting, so the order parameter is included. sym_right = kwant.TranslationalSymmetry((a, 0)) lead1 = kwant.Builder(sym_right) lead1[(lat(0, j) for j in range(W))] = (4 * t - mu + E_z) * tau_z + Delta * tau_x lead1[kwant.builder.HoppingKind((1, 0), lat, lat)] = -t * tau_z + 1j * alpha * tau_y lead1[kwant.builder.HoppingKind((0, 1), lat, lat)] = -t * tau_z - 1j * alpha * tau_x #### Attach the leads and return the system. #### syst.attach_lead(lead0) syst.attach_lead(lead1) for i in syst.leads[0].interface: for j in syst.neighbors(i): syst[i, j] = 0.1 * tau_0 for i in syst.leads[1].interface: for j in syst.neighbors(i): syst[i, j] = 0.1 * tau_0 return syst def plot_conductance(syst, energies): # Compute conductance data = [] for energy in energies: smatrix = kwant.smatrix(syst, energy) # Conductance is N - R_ee + R_he data.append( smatrix.submatrix((0, 0), (0, 0)).shape[0] - smatrix.transmission((0, 0), (0, 0)) + smatrix.transmission((0, 1), (0, 0)) ) pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") pyplot.show() def check_PHS(syst): # Scattering matrix s = kwant.smatrix(syst, energy=0) # Electron to electron block s_ee = s.submatrix((0, 0), (0, 0)) # Hole to hole block s_hh = s.submatrix((0, 1), (0, 1)) print("s_ee: \n", np.round(s_ee, 3)) print("s_hh: \n", np.round(s_hh[::-1, ::-1], 3)) print("s_ee - s_hh^*: \n", np.round(s_ee - s_hh[::-1, ::-1].conj(), 3), "\n" ) # Electron to hole block s_he = s.submatrix((0, 1), (0, 0)) # Hole to electron block s_eh = s.submatrix((0, 0), (0, 1)) print("s_he: \n", np.round(s_he, 3)) print("s_eh: \n", np.round(s_eh[::-1, ::-1], 3)) print("s_he + s_eh^*: \n", np.round(s_he + s_eh[::-1, ::-1].conj(), 3)) def main(): syst = make_system(W=10) # Check that the system looks as intended. kwant.plot(syst) # Finalize the system. syst = syst.finalized() # Check particle-hole symmetry of the scattering matrix check_PHS(syst) # Compute and plot the conductance plot_conductance(syst, energies=[0.002 * i for i in range(-10, 100)]) # Call the main function if the script gets executed (as opposed to imported). # See <http://docs.python.org/library/__main__.html>. if __name__ == "__main__": main()

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Unit of Energy in conductance vs Energy plots
by sahu.ajit92＠gmail.com 09 Sep '22

09 Sep '22

Dear all, 1. In the example of KWANT, there is a conductance vs Energy plot. In this plot, the unit of Energy is mentioned as [t]. what does t signify? is it in eV? 2. In the band structure of zig-zag graphene lead, there are upper and lower parts in the band structure. which one is the conduction band? I'm attaching a figure for 2nd question. figure https://drive.google.com/file/d/1dmhZkSOWu4ERvMfP855vgbYpAyt5NTcn/view?usp=…

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Re: Unit of Energy in conductance vs Energy plots
by Ajit Kumar Sahu 09 Sep '22

09 Sep '22

Thank you sir. Cleared all my doubts.

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How to realize local Chern marker in kwant.kpm?
by Jerry xhm 08 Sep '22

08 Sep '22

Hello Kwant community, I want to calculate the local Chern marker in kwant, roughly a real-space approach to topological numbers such as the Chern number. Basically, one needs to calculate the expectation value of the commutator [x, y] of position operators projected to the occupied states. The main reference is Eqs.(6-9) in [ http://dx.doi.org/10.1103/PhysRevB.84.241106 ], which has been applied in many other cases. My understanding is that kwant can define x,y as ‘density operators’ and then this task should fit into the kwant.kpm.correlator framework, because it is a correlation function at zero temperature between the two operators. So I tried in the following code a few ways (some commented) for the simplest square-lattice quantum Hall effect (QHE). But it does not seem to give a sign of quantization, although I'm sure that the same system gives reasonable QHE in conductivity and conductance calculation. E.g., for 1/B in [3, 12], it should show the C=1, C=2 plateaus. I feel that I probably missed or misunderstood something basic here. Is my use of operator and kpm.correlator correct here? Thank you for your help! from cmath import exp import numpy as np import matplotlib.pyplot as plt import kwant import scipy.sparse.linalg as sla import scipy.sparse as sparse import tinyarray Nx, Ny = 101, 101 Lx, Ly = Nx-1, Ny-1 edge=18 eps = 1e-7 L_sys = [Lx, Ly] Ef=0.4 def make_system(a=1, t=1): lat = kwant.lattice.square(a, norbs=1) syst = kwant.Builder() def fluxx(site1, site2, Bvec): pos = [(site1.pos[i]+site2.pos[i]-L_sys[i])/2.0 for i in range(2)] return exp(-1j * Bvec * pos[1] ) def fluxy(site1, site2, Bvec): pos = [(site1.pos[i]+site2.pos[i]-L_sys[i])/2.0 for i in range(2)] return 1 #exp(-1j * (-Bvec * pos[0])) def hopx(site1, site2, Bvec): # The magnetic field is controlled by the parameter Bvec return fluxx(site1, site2, Bvec) * t def hopy(site1, site2, Bvec): # The magnetic field is controlled by the parameter Bvec return fluxy(site1, site2, Bvec) * t def onsite(site): return 4*t #### Define the scattering region. #### syst[(lat(x, y) for x in range(L_sys[0]) for y in range(L_sys[1]))] = onsite # hoppings in x-direction syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = hopx # hoppings in y-directions syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = hopy # Finalize the system. fsyst = syst.finalized() return lat, fsyst lat, fsyst = make_system() sites = fsyst.sites X = sparse.diags([site.pos[0] for site in sites]) Y = sparse.diags([site.pos[1] for site in sites]) def where(site): pos=site.pos x0_min, y0_min = edge-eps, edge-eps x0_max, y0_max= Lx-edge+eps, Ly-edge+eps return x0_min < pos[0] < x0_max and y0_min < pos[1] < y0_max def op0(site): return site.pos[0] def op1(site): return site.pos[1] def chern(lat, fsyst): # correlator=kwant.kpm.Correlator(fsyst, params=params, operator1=kwant.operator.Density(fsyst, onsite=op0, where=where),\ # operator2=kwant.operator.Density(fsyst, onsite=op1, where=where),\ # num_vectors=10, num_moments=400, vector_factory=kwant.kpm.RandomVectors(fsyst, where=where),\ # bounds=(0, 8.), eps=0.01, rng=0, kernel=None, mean=True, accumulate_vectors=False) correlator=kwant.kpm.Correlator(fsyst, params=params, operator1=kwant.operator.Density(fsyst, onsite=op0),\ operator2=kwant.operator.Density(fsyst, onsite=op1),\ num_vectors=10, num_moments=400, vector_factory=kwant.kpm.RandomVectors(fsyst, where=where),\ bounds=(0, 8.), eps=0.01, rng=0, mean=True, accumulate_vectors=False) # correlator=kwant.kpm.Correlator(fsyst, params=params, operator1=X,\ # operator2=Y,\ # num_vectors=10, num_moments=400, vector_factory=kwant.kpm.RandomVectors(fsyst, where=where),\ # bounds=(0, 8.), eps=0.01, rng=0, kernel=None, mean=True, accumulate_vectors=False) c = correlator(mu=Ef, temperature=0.0) / ((Lx-2*edge)*(Ly-2*edge)) print(2*np.pi*c) print("Chern number: ", np.real(np.pi*1j*(c-c.conj()))) # Chern number Bfields = [] for BInv in np.arange(3., 12., 2): Bfields.append(1/BInv) NB = len(Bfields) params = dict(Bvec=0) for B in Bfields: params['Bvec'] = B print("B: ", B) chern(lat, fsyst)

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