Mailman 3 September 2021 - Kwant-discuss

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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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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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

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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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Graphene Conductor-Superconductor Conductance above Gap
by Henry Axt 21 Feb '22

21 Feb '22

Hello, I set up a graphene lattice with periodic boundary conditions that is connected to a superconductor (simulated with an electron and hole lattice). When observing the transmission between the electron and hole lattice, I see a small amount conductance above the superconducting gap. I do not see this with something like a square lattice. What could explain this?

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Graphene Sparse vs Dense - Energy calculation values dependent on graphene length
by Henry Axt 06 Oct '21

06 Oct '21

This is related to my other post, but as I see them as different (maybe not?) problems, I will post a second thread. The situation is, again, a graphene strip with periodic boundary conditions in one direction and open boundaries in the other. Using the sparse solver from linalg, I have solved for energies and gotten reasonable results for the energies, except for the case when the length (along the axis of the open boundary condition) is metallic. Attached is a code that shows it for length = 17, which is a metallic length for graphene (has energies at zero), but it does not produce messy energy calculation for when it is not a metallic length, i.e. something like length = 16. """ Graphene wire with periodic boundary conditions in the vertical direction. """ import kwant from math import pi, sqrt, tanh, cos, ceil, floor, atan, acos, asin from cmath import exp import numpy as np import scipy import scipy.linalg as lina sin_30, cos_30, tan_30 = (1 / 2, sqrt(3) / 2, 1 / sqrt(3)) def create_closed_system(length, width, lattice_spacing, onsite_potential, hopping_parameter, boundary_hopping): padding = 0.5*lattice_spacing*tan_30 def calc_total_length(length): total_length = length N = total_length//lattice_spacing # Number of times a graphene hexagon fits in horizontially fully new_length = N*lattice_spacing + lattice_spacing*0.5 diff = total_length - new_length if diff != 0: length = length - diff total_length = new_length return total_length def calc_width(width,lattice_spacing, padding): stacking_width = lattice_spacing*((tan_30/2)+(1/(2*cos_30))) N = width//stacking_width if N % 2 == 0.0: # Making sure that N is odd. N = N-1 new_width = N*stacking_width + padding width = new_width return width, int(N) def rectangle(pos): x,y = pos if (0 < x <= total_length) and (-padding <= y <= width -padding): return True return False def lead_shape(pos): x, y = pos return 0 - padding <= y <= width def tag_site_calc(x): return int(-1*(x*0.5+0.5)) #Initation of geometrical limits of the lattice of the system total_length = calc_total_length(length) width, N = calc_width(width, lattice_spacing, padding) # The definition of the potential over the entire system def potential_e(site,pot): return pot ### Definig the lattices ### graphene_e = kwant.lattice.honeycomb(a=lattice_spacing,name='e') a, b = graphene_e.sublattices sys = kwant.Builder() # The following functions are required for input in the def onsite_shift_e(site, pot): return potential_e(site,pot) sys[graphene_e.shape(rectangle, (0.5*lattice_spacing, 0))] = onsite_shift_e sys[graphene_e.neighbors()] = -hopping_parameter ### Boundary conditions for scattering region ### for site in sys.sites(): (x,y) = site.tag if float(site.pos[1]) < 0: if str(site.family) == "<Monatomic lattice e1>": sys[b(x,y),a(int(x+tag_site_calc(N)),N)] = -boundary_hopping kwant.plot(sys) return sys def eigen_vectors_and_values(sys,sparse_dense,k): #sparse = 0, dense = 1 if sparse_dense == 0: ham_mat = sys.hamiltonian_submatrix(params=dict(pot=0.0), sparse=True) eigen_val, eigen_vec = scipy.sparse.linalg.eigsh(ham_mat.tocsc(), k=k, sigma=0, return_eigenvectors=True) if sparse_dense == 1: ham_mat = sys.hamiltonian_submatrix(params=dict(pot=0.0), sparse=False) eigen_val, eigen_vec = lina.eigh(ham_mat) #sort the ee and ev in ascending order idx = eigen_val.argsort() eigen_val= eigen_val[idx] eigen_vec = eigen_vec[:,idx] if sparse_dense == 1: eigen_val = eigen_val[int(len(eigen_val)*0.5-k*0.5):] # Remove first part eigen_val = eigen_val[:k] # Take first k-values eigen_vec = eigen_vec[:,int(len(eigen_val)*0.5-k*0.5):] eigen_vec = eigen_vec[:,:k] return eigen_vec, eigen_val def main(): sys = create_closed_system(length = 17.0, width = 20.0, lattice_spacing = 1.0, onsite_potential = 0.0, hopping_parameter = 1.0, boundary_hopping = 1.0) sys = sys.finalized() #sparse = 0, dense = 1 eigen_vec, eigen_val = eigen_vectors_and_values(sys,sparse_dense=0,k=50) eigen_vec2, eigen_val2 = eigen_vectors_and_values(sys,sparse_dense=1,k=50) print("Eigenenergy difference (between sparse and dense): ") for i_ in range(len(eigen_val)): print(eigen_val[i_] - eigen_val2[i_] ) return if __name__ == "__main__": main()

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Conductivity cannot get quantized result
by Jerry xhm 29 Sep '21

29 Sep '21

Hello Kwant community, I have played with a Weyl semimetal system for a while, using Kwant's conductance features (with leads). And I do get the quantized Hall response under magnetic field as reported in this work [PRL 125, 036602]. Now with exactly the same system (size, Fermi energy, hopping, etc.), I remove the leads and try the kpm.conductivity features. However, there is no sign of quantization at all, as far as I've tried with LocalVectors or RandomVectors and tuning num_moments or so. To make sure of my use of kpm.conductivity, I tried another known unquantized Hall effect (sigma_xy) in this system without magnetic field and it looked not unreasonable; I also changed the system to a simple 2D QHE one and the same code worked well to see quantization. This seems rather unclear to me. Could you please help with how should I correctly calculate the conductivity in Kwant here? The code below plots sigma_xz and sigma_zx with respect to a few inverse magnetic fields. Thanks! from cmath import exp from math import cos, sin, pi import numpy as np import time import sys import kwant # For plotting from matplotlib import pyplot as plt # For matrix support import tinyarray # define Pauli-matrices for convenience 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]]) Lx=25 Ly=20 Lz=25 D=0.02 D1=D D2=4*D A=1 M=0.3 kw=pi/2.0 E_F=2*D2*(1-cos(kw)) # Fermi energy norbs=2 def make_system(a=1, t=1): lat = kwant.lattice.cubic(a, norbs=norbs) syst = kwant.Builder() def fluxx(site1, site2, B): pos = [(site1.pos[i]+site2.pos[i])/2.0 for i in range(3)] return exp(-1j * (B * pos[2])) def hopx(site1, site2, B): return fluxx(site1, site2, B) * (-D2 * sigma_0 + 1j * A/2 * sigma_x + M * sigma_z) def hopy(site1, site2): return (-D1 * sigma_0 + 1j * A/2 * sigma_y + M * sigma_z) def hopz(site1, site2): return (-D2 * sigma_0 + M * sigma_z) def onsite(site): return (2 * D1 + 2 * D2 * 2) * sigma_0 + 2 * M * ((1 - cos(kw)) - 3) * sigma_z syst[(lat(x, y, z) for x in range(Lx) for y in range(Ly) for z in range(Lz))] = onsite # hoppings in x-direction syst[kwant.builder.HoppingKind((1, 0, 0), lat, lat)] = hopx # hoppings in y-directions syst[kwant.builder.HoppingKind((0, 1, 0), lat, lat)] = hopy # hoppings in z-directions syst[kwant.builder.HoppingKind((0, 0, 1), lat, lat)] = hopz return lat, syst.finalized() def plot_curves(labels_to_data): plt.figure(figsize=(12,10)) for label, (x, y) in labels_to_data: plt.plot(x, y, label=label, linewidth=2) plt.legend(loc=2, framealpha=0.5) plt.xlabel("1/B") plt.ylabel(r'$\sigma$') plt.show() plt.clf() def cond(lat, syst, random_vecs_flag=False): center_width = min(Lx,Ly,Lz)//20 center_pos = [Lx//2,Ly//2,Lz//2] def center(s): return (np.abs(s.pos[0]-center_pos[0]) < center_width) and (np.abs(s.pos[1]-center_pos[1]) < center_width) and (np.abs(s.pos[2]-center_pos[2]) < center_width) num_mmts = 200; if random_vecs_flag: #use RandomVecs center_vecs = kwant.kpm.RandomVectors(syst, where=center) one_vec = next(center_vecs) num_vecs = 20 else: # use LocalVecs center_vecs = np.array(list(kwant.kpm.LocalVectors(syst, where=center))) one_vec = center_vecs[0] num_vecs = len(center_vecs) area_per_orb = np.abs(np.linalg.det(lat.prim_vecs)) norm = area_per_orb * np.linalg.norm(one_vec) ** 2 / norbs Bfields = [] Binvs = np.arange(2.4, 65.2, 30.5) for BInv in Binvs: Bfields.append(1/BInv) params = dict(B=0) cond_array_zx = [] cond_array_xz = [] for B in Bfields: params['B'] = B cond_zx = kwant.kpm.conductivity(syst, params=params, alpha='z', beta='x', mean=True, rng=0, num_moments=num_mmts, num_vectors=num_vecs, vector_factory=center_vecs) cond_xz = kwant.kpm.conductivity(syst, params=params, alpha='x', beta='z', mean=True, rng=0, num_moments=num_mmts, num_vectors=num_vecs, vector_factory=center_vecs) cond_array_zx.append(cond_zx(mu=E_F, temperature=0.0)/ norm) cond_array_xz.append(cond_xz(mu=E_F, temperature=0.0)/ norm) cond_array_zx = np.array(cond_array_zx) * Ly # 3D conductivity * Ly gives the sheet 2D-like conductivity supposed to be quantized cond_array_xz = np.array(cond_array_xz) * Ly print(cond_array_zx) print(cond_array_xz) print('Calculating duration (s) until finalising system: ', time.time() - global_start_time) plot_curves([(r'$\sigma_{zx}$', (Binvs, cond_array_zx)), (r'$\sigma_{xz}$', (Binvs, cond_array_xz))]) def main(): lat, syst = make_system() # Check that the system looks as intended. #kwant.plot(syst, file="lattice_fig.pdf") cond(lat, syst) if __name__ == '__main__': global_start_time = time.time() main()

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Graphene Eigenenergies and Eigenstate calculations via Sparse vs Dense
by Henry Axt 28 Sep '21

28 Sep '21

Hello, I am not too sure where my problem lies so, I will post my question here, since I might suspect it has something to do with the kwant interface. I have a simple graphene strip with periodic boundaries in the vertical direction and open boundaries in the other direction. To calculate the energies and states of this system, I feed the hamiltonian from kwant into either the dense or the sparse solver from linalg. I do not understand why I am getting significant difference in the values of the eigenstates using sparse vs using density, especially considering the small size of my system. Attached is a code that reproduces the problem: """ Graphene wire with periodic boundary conditions in the vertical direction. """ import kwant from math import pi, sqrt, tanh, cos, ceil, floor, atan, acos, asin from cmath import exp import numpy as np import scipy import scipy.linalg as lina sin_30, cos_30, tan_30 = (1 / 2, sqrt(3) / 2, 1 / sqrt(3)) def create_closed_system(length, width, lattice_spacing, onsite_potential, hopping_parameter, boundary_hopping): padding = 0.5*lattice_spacing*tan_30 def calc_total_length(length): total_length = length N = total_length//lattice_spacing # Number of times a graphene hexagon fits in horizontially fully new_length = N*lattice_spacing + lattice_spacing*0.5 diff = total_length - new_length if diff != 0: length = length - diff total_length = new_length return total_length def calc_width(width,lattice_spacing, padding): stacking_width = lattice_spacing*((tan_30/2)+(1/(2*cos_30))) N = width//stacking_width if N % 2 == 0.0: # Making sure that N is odd. N = N-1 new_width = N*stacking_width + padding width = new_width return width, int(N) def rectangle(pos): x,y = pos if (0 < x <= total_length) and (-padding <= y <= width -padding): return True return False def lead_shape(pos): x, y = pos return 0 - padding <= y <= width def tag_site_calc(x): return int(-1*(x*0.5+0.5)) #Initation of geometrical limits of the lattice of the system total_length = calc_total_length(length) width, N = calc_width(width, lattice_spacing, padding) # The definition of the potential over the entire system def potential_e(site,pot): return pot ### Definig the lattices ### graphene_e = kwant.lattice.honeycomb(a=lattice_spacing,name='e') a, b = graphene_e.sublattices sys = kwant.Builder() # The following functions are required for input in the def onsite_shift_e(site, pot): return potential_e(site,pot) sys[graphene_e.shape(rectangle, (0.5*lattice_spacing, 0))] = onsite_shift_e sys[graphene_e.neighbors()] = -hopping_parameter ### Boundary conditions for scattering region ### for site in sys.sites(): (x,y) = site.tag if float(site.pos[1]) < 0: if str(site.family) == "<Monatomic lattice e1>": sys[b(x,y),a(int(x+tag_site_calc(N)),N)] = -boundary_hopping kwant.plot(sys) return sys def eigen_vectors_and_values(sys,sparse_dense,k): #sparse = 0, dense = 1 if sparse_dense == 0: ham_mat = sys.hamiltonian_submatrix(params=dict(pot=0.0), sparse=True) eigen_val, eigen_vec = scipy.sparse.linalg.eigsh(ham_mat.tocsc(), k=k, sigma=0, return_eigenvectors=True) if sparse_dense == 1: ham_mat = sys.hamiltonian_submatrix(params=dict(pot=0.0), sparse=False) eigen_val, eigen_vec = lina.eigh(ham_mat) #sort the ee and ev in ascending order idx = eigen_val.argsort() eigen_val= eigen_val[idx] eigen_vec = eigen_vec[:,idx] if sparse_dense == 1: eigen_val = eigen_val[int(len(eigen_val)*0.5-k*0.5):] # Remove first part eigen_val = eigen_val[:k] # Take first k-values eigen_vec = eigen_vec[:,int(len(eigen_val)*0.5-k*0.5):] eigen_vec = eigen_vec[:,:k] return eigen_vec, eigen_val def main(): sys = create_closed_system(length = 16.0, width = 20.0, lattice_spacing = 1.0, onsite_potential = 0.0, hopping_parameter = 1.0, boundary_hopping = 1.0) sys = sys.finalized() #sparse = 0, dense = 1 eigen_vec, eigen_val = eigen_vectors_and_values(sys,sparse_dense=0,k=50) eigen_vec2, eigen_val2 = eigen_vectors_and_values(sys,sparse_dense=1,k=50) print("Eigenenergy difference: ") for i_ in range(len(eigen_val)): print(eigen_val[i_] - eigen_val2[i_] ) new_m = eigen_vec - eigen_vec2 new_m = abs(new_m) print("Max difference in eigenvector values: ", np.amax(new_m)) return new_m if __name__ == "__main__": new_m = main()

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Calculate conductivity of 3D periodic boundary condition system with disorder
by Jerry xhm 26 Sep '21

26 Sep '21

Hello, I am thinking of calculating the conductivity with Kwant's KPM method of a 3D system with disorder and periodic boundary conditions (PBC) in every direction (no leads). Reading some old posts, it seems to be some problem with multiple translational symmetries (or PBCs?). In my case, because of adding disorder potential, I only want PBC rather than translational symmetry. So I presume the wraparound module is irrelevant here. Is that possible and if any problem when further calculating its conductivity? Naively we only need to manually add hoppings that 'wrap around' the lattice. Does this cause any problem in Kwant and is there any good practice/trick for this? Thank you!

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Kwant-discuss September 2021