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
# Tutorial 2.3.1. Matrix structure of on-site and hopping elements # ================================================================ # # Physics background # ------------------ # Gaps in quantum wires with spin-orbit coupling and Zeeman splititng, # as theoretically predicted in # http://prl.aps.org/abstract/PRL/v90/i25/e256601 # and (supposedly) experimentally oberved in # http://www.nature.com/nphys/journal/v6/n5/abs/nphys1626.html # # Kwant features highlighted # -------------------------- # - Numpy matrices as values in Builder # Modified by Nathaniel Seil on 24/7/2020 to simulate 1000 x 30 array of basic atoms. import kwant # For plotting from matplotlib import pyplot # For matrix support import tinyarray import numpy as np # define Pauli-matrices for convenience hbar = 1.054571628E-34 # J s sigma_0 = hbar/2 * tinyarray.array([[1, 0], [0, 1]]) sigma_x = hbar/2 * tinyarray.array([[0, 1], [1, 0]]) sigma_y = hbar/2 * tinyarray.array([[0, -1j], [1j, 0]]) sigma_z = hbar/2 * tinyarray.array([[1, 0], [0, -1]]) def make_system(t=1.0, alpha=0.5, e_z=0.0, W=10, L=30): # default e_z = 0.08 # Start with an empty tight-binding system and a single square lattice. # `a` is the lattice constant (by default set to 1 for simplicity). a = 5E-10 #metres lat = kwant.lattice.square(a) syst = kwant.Builder() #### Define the scattering region. #### syst[(lat(x, y) for x in range(L) for y in range(W))] = \ 4 * t * sigma_0 + e_z * sigma_z # above is non-derivative components. syst[kwant.builder.HoppingKind((1, 0), lat, lat)] = \ -t * sigma_0 + 1j * alpha * sigma_y / 2 # hoppings in y-directions syst[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_x / 2 #### Define the left lead. #### lead = kwant.Builder(kwant.TranslationalSymmetry((-1*a, 0))) lead[(lat(0, j) for j in range(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 / 2 # hoppings in y-directions lead[kwant.builder.HoppingKind((0, 1), lat, lat)] = \ -t * sigma_0 - 1j * alpha * sigma_x / 2 #### Attach the leads and return the finalized system. #### syst.attach_lead(lead) syst.attach_lead(lead.reversed()) return syst def plot_conductance(syst, energies): # Compute conductance data = [] for energy in energies: smatrix = kwant.smatrix(syst, energy) data.append(smatrix.transmission(1, 0)) pyplot.figure() pyplot.plot(energies, data) pyplot.xlabel("energy [t]") pyplot.ylabel("conductance [e^2/h]") # pyplot.show() pyplot.savefig("author_demo_conductance.png") def main(): a = 5E-10 # metres hbar = 1.054571628E-34 # J s m_electron = 9.109E-31 # kg B = 0.001 # Teslas c = 2.99792458E+8 # m/s echarge = 1.602E-19 # C e_z = 2.0023/2 * hbar * echarge * B / (m_electron * c) # Hydrogen W = 30 L = 1000 alpha = 0.5 t = (hbar ** 2)/(2*m_electron*(a**2)) syst = make_system(t, alpha, e_z, W, L) # Check that the system looks as intended. kwant.plot(syst) # Finalize the system. syst = syst.finalized() # We should see non-monotonic conductance steps. energies=[0.01 * i*t - 0.3*t for i in range(100)] energies = np.array(energies) voltages = energies/echarge print(voltages) plot_conductance(syst, energies=energies) # 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()
